On the theories of Morse and Lusternik-Schnirelman for open bounded sets on Fredholm Hilbert manifolds

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPILCATIONS 50, 80-107 (1975)
On the Theories of Morse and Lusternik-Schnirelman for Open Bounded Sets on
Fredholm Hilbert Manifolds
E. H. ROTHE
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48104
Submitted by C. L. Dolph
1. INTRODUCTION
The Morse theory of critical points of a real valued functionf defined on a finite dimensional manifold M without boundary was generalized by Palais and Smale to the case where h4 is a Hilbert manifold without boundary [8, lo]. In particular if all critical points are nondegenerate (and therefore isolated) the well known Morse inequalities between the Betti number R, of M and the Morse numbers M, were generalized in an appropriate form (see [8, p. 338, Theorem 71; for the definition of M, see Eq. (3.10) of the present paper; see also Remark at the end of Section 3).
On the other hand the Morse theory for real valued functions f defined in the closure r of a bounded open set V in a finite dimensional vector space E was generalized to the case where E is a Hilbert space. The case where f satisfies a “regular boundary condition” (stating essentially that at every point of the boundary p of V the gradient off is exteriorly directed, see Assumption 2.4) was treated in [14], and the case of “general boundary conditions” (where f is allowed to have the direction of the interior normal at a finite number of points of p) was treated in [13]. In either case the statement of the boundary condition required the existence of a unique exterior unit normal, and it was this requirement which motivated the assumption that p be a Fredholm manifolds. In the case of the regular boundary condition sufficient conditions for the validity of the Morse relations were given, (see [14, Theorem 81 where f is supposed to be bounded).
The present paper aims at a synthesis of the investigations mentioned in the preceding paragraph with those referred to in the first paragraph by treating the case where the domain off is the closure r of an open bounded connected subset V of a Hilbert-Fredholm-Riemannian manifold M and where a regular boundary condition is satisfied. (Thus critical points on the boundary p of V are excluded, and so are corners on l? Such points are
80 Copyright 0 1975 by Academic Presr, Inc. AU rights of reproduction in any form reserved.

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 81
admitted in the investigations of D. Braess concerning the finite dimensional case (see [l]).
Section 2 below is mainly concerned with the geometric background. Under the assumption that the Hilbert manifold M is Fredholm the notion of a hyper-submanifold N of M is introduced in analogy to the notion of an (n - 1)-dimensional submanifold of an n-dimensional manifold (Definition 2.2). It is supposed that the boundary a of the open subset V of M is a hyper-submanifold of M and that M is Riemannian. Then a unique exterior unit normal to p can be defined (Definition 2.3) and the regular boundary condition (Assumption 2.4) can be stated. This assumption together with the other basic assumptions of this paper (Assumptions 2.1-2.3) allow us to show that a “gradient line” through a point of I’ does not intersect the boundary p if followed in the sense of decreasing f (Theorem 2.2).
Section 3 deals with the Morse theory. First a function f defined on a topological space S is considered and under rather general assumptions (Assumption 3.1) critical groups C,(c) are attached to a critical value c off. Sufficient conditions for the validity of the Morse relations (3.11) and (3.12) (which latter are written in terms of the ranks of the groups C,(c)) are given (Theorem 3.1 whose proof is essentially the same as the one given by Pitcher [I l] in the finite dimensional case). It is then shown that these sufficient conditions are satisfied if S = r and if Assumption 3.2 is added to Assump- tions 2.1-2.4 (Theorem 3.2).
So far only critical levels were considered. The consideration of critical points begins with Definition 3.4. Critical groups are attached to critical points and a relation is established between the group C,(c) and the groups of the critical points at level c provided there are only a finite number of critical points at that level (Theorem 3.3). This finiteness condition is (on account of Lemma 2.8) certainly satisfied if there are althogether only iso- lated (not necessarily nondegenerate) critical points. Thus in this case all groups C,(c) are finitely generated and therefore the Morse relations (3.12) hold. For the special case that all critical points are non-degenerate see the Remark following Theorem 3.4.
In Section 4 it is shown that for functions which are bounded from below the main facts of the Lusternik-Schnirelman theory hold under Assumptions 2.1-2.4. The proof consists in verifying that under these assumptions a set of conditions is satisfied which were proved to be sufficient by F. Browder [2, Theorems 2 and 31. A different proof for assertion (ii) of Theorem 4.1 can be given by generalizing a method employed by Seifert and Threlfall [16, p. 911 in the case of a finite dimensional manifold. This proof requires more assumptions on f but it is more constructive in that it constructs R closed sets covering v if there are K critical points by extending the “cylin- drical neighborhood” of each critical point. (For the definition of a cylindrical

82 E. H. RoTHE
neighborhood see [16, Section 91 in the finite dimensional case, and [14, Section 51 in the Hilbert space case.) Details of this proof will be given in another paper [15a].
For a short survey of the history of the Lusternik-Schnirelman theory we refer the reader to [2, pp. 5 and 61.
We conclude this introduction by listing a few notations used in the sequel: If F is a real valued function with domain S and a area1 number then
{f=u} ={xESIf(x) =a>; fa =+ESIf(X) <a>, and Ja = (X E S 1 f(x) < a}.
In general an upper bar denotes closure. The distance of the elements x and y in a metric space will be denoted by [I x, y I( while B(x, a) denotes the open ball with center x and radius a. If b > a the [a, b] denotes the closed interval with endpoints a and b while the point set [a, b] - (b} is denoted by [a, b). The symbol “M” between two groups denotes isomorphism. The zero element of a vectorspace will be denoted by 0.
2. THE GEOMETRICAL BACKGROUND
Let M be a connected CT manifold without boundary modelled on a fixed Hilbert space E(r is a positive integer). For the definition of such a manifold as well as for the definition and properties of charts and of an atlas for M we refer the reader to [7] or [8]. Here we recall that a chart for M at a point x0 E M is a pair (U, 9) where U is an open neighborhood of x0 and 4 is a bijection of U onto an open subset of E, and that an atlas A for M is a col- lection of charts such that the neighborhoods U cover M, with the additional property: if (U, 4) and ( W, (CI) are two charts in A for which the intersection U n W is not empty then the map
+p:$(Un W)+#(Un W) (2.1)
is a CT isomorphism, i.e., a one to one map onto admitting continuous (Frechet) differentials up to and including order r.
Since r > 1 it follows that the differential d#$-l(~O , u) of the map (2.1) at a point u,, E $( U) with “increment” u E E is (as function of U) a (bounded) linear one to one map of E onto E.
DEFINITION 2.1. M is called a Fredholm manifold if there exists an atlas A for M such that for any two charts (U, 4) and ( W, #) in A

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMA 83
where C is completely continuous. (See [4].) Concerning the question when a given manifold can be “reduced” to a Fredholm manifold we refer the reader to [4, p. 751 and [3, p. 7681. From the statements made there it follows, e.g., that every paracompact manifold modelled on a separable Hilbert space can be so reduced.
Out next goal is to give a definition of a hypersubmanifold of a Fredholm manifold M. From the definition of a submanifold N of M as given in [7, Chapter II, Section 21, and adopted in the present paper, the following fact follows directly; if y E N then there exists a chart (U, 4) for M at y with the property: there exists a direct decomposition
E = F + E2, (2.3)
of E into two closed (linear) subspaces, and two sets Vi and Va which are open subsets of El and E2 resp. such that
and
c+(U) = v1 x v’2, (2.4)
+(Un N) = V2CE2.
A chart (W, #) for iV at y is obtained by setting
(2.5)
W=Ur\N, #=(blw (2.6)
and the charts so obtained form an atlas for N. We now would like to give the following
DEFINITION 2.2. The submanifold N of the Fredholm manifold M is said to be hypersubmanifold of M at the point y E N if the space El in the decomposition (2.3) is one dimensional. N is called a hypersubmanifold of M if it is a hypersubmanifold at everyone of its points.
However to make this definition legitimate we obviously have to prove the following.
LEMMA 2.1. Let (W, #) be a chartfm Nat y as in (2.6), and let (#‘, $) be another chart for N at y obtained from a chart ( o,& for M at y for which there exist a direct decomposition
E = F $ i?, (2.3)
and sets p which are open subsets of l? (i = 1,2) szrch that & 0) = p1 x r2 and $( 0 n N) = Vz C e2. It is asserted: if El is one dimensional then 81 is onedimensional.

84 E. H. ROTHE
The proof is based on the following.
LEMMA 2.2. If E2 is hyperspace in E (i.e., a closed linear subspace of codimension 1) then the image of E2 under a nonsingular linear map E -+ E of the form u + C(u) with a completely continuous C is also a hyperspace in E.
This lemma was proved in [13, Lemma 3.21.
Proof of Lemma 2.1. We consider maps &b-l: +( U n 0) -+-I$( U n a), &,b-1: $(Wn tv)-Hp(Wn tv), and, at a point ur E #(W n @), their differentials
L(u) = d&W, , u>, l(u) = d$,F(u, , u).
Then Z(U) is a linear map of E2 onto J!?, whileL(u) maps E onto E. Now it is easily verified that z&k’ is a restriction of &-’ from +( U n U), to fj( W n IP). It is not hard to see (using the definition of a FrCchet differential) that this fact implies that I(u) is the restriction of L(u) from E to E2. Therefore L(E2) = Z(E2) = B2. Application of Lemma 2.2 now finishes the proof of Lemma 2.1.
Later on we will deal with real valued functions defined on the closure r of an open bounded subset V of M. From now on we will assume that M is a Fredholm manifold and that the boundary p of V is a hypersubmanifold of M (cf. Assumption 2.1 below).
The following lemma is a consequence of this assumption.
LEMMA 2.3. Let (U, 4) be a chart for iI4 at a point y of V with 4(y) = 8, and let El and E2 be as in (2.3) with N = V. Then there exist a unit vector el E 6 and a positive number [,-, of the following property: if 0 < t: < CO and af the sets B,+ and B,- are defined by
B,+ = (u E B(4 r> I (u, el> > O),
therz (i) the sets 4-l(BE+) and +-l(B,-) are contained in U, and (ii) the points of +-l(Bt+) are exterior to V while @l(B,-) C V.
Proof. Assertion (i) follows trivially from the fact that 4(U) is open. Let us then prove the iirst part of assertion (ii) with a positive 5 < to where 5s satisfies (i). $-l(B(8, b)) is a neighborhood of the boundary pointy of V and therefore contains a point yO exterior to V. Let uO = $(y,,). Then ‘co E+(U) = Vl x V, C E, and if P is one of the two unit vectors which span El we have by (2.3) the representation
t1 = (240 , Zl), e2e V2CEa. P-8)

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 85
Here t, # 0 since otherwise
y. = +-‘(uo) = +-l(3) c 4-y V2) c P,
in contradiction to the fact the y0 is an exterior point. We now define e1 by setting er = P if tr is positive, and e = --P if tr is negative. Then (u,,el) > 0.
With e1 thus defined we will prove that (qb-l(u,) is exterior to V for u1 in the set B,+ given by (2.7). Indeed uor = (1 - a) u,, + olur E B,+ for 0 < OL < 1 since B,+ is convex. In particular u, has no point in common with E2. From this it follows that $-l(uU) is a continuous curve not intersecting $‘. Since (5-1(u0) = y0 is an exterior point so is $-‘(ur). This finishes the proof of the first part of assertion (ii), and the second part is proved correspondingly.
COROLLARY TO LEMMA 2.3. Every point x0 in V has a nezghborhood (with respect to r) which is contractible to x0 on v. (This corollary will be used in Section 4).
Proof. Consider first a point x,, = y E p. Then in the notation of the preceding lemma it follows from that lemma that for 5 small enough the set +-l(B& is a neighborhood of x,, with respect to p. This neighborhood satisfies the requirement of the corollary since Bc is contractible on itself to x0 . The proof is still simpler if x0 E V.
We now recall certain facts concerning tangent spaces to M and N. Our goal is to arrive at a definition of exterior normals at points of 8.
Let ( Uj , +i) be the charts at a point of M indexed by an index set I, and let uj denote points in E. It can be verified that an equality of the form
uk = d$k+il(% 1 ui), uo = ~dxo), xoEUjn u,, (2.9)
represents an equivalence relation. If (2.9) holds we say that the triples (U3.,h,ud and (Ukr+k, uk) are equivalent or simply that uj and z+ are equivalent (with respect to the above charts), in symbols: r+ N uk .
The tangentspace to M at x, denoted by M, , is then defined as the set of equivalence classes under the relation (2.9). A point t of M, is a collection {ui}jsl of equivalent points. With a natural definition of addition and multi- plication by a scalar, Mz becomes a linear space.
We recall the definition of the differential of a map F of M into another manifold M* modelled on a Hilbert space E*. Let (U, 4) be a chart at x0 E M, and let (U*, $*) be a chart at y. = F(x,). Let Fbern be the map of b(U) on $*( U*) defined by Fm.6 = +*F+-l. Then dF,&u, , U) is well defined if uog$(U) and uEE. If (o,$) is another chart of x0 it can be shown that
dF,&,; 4 = dF,,,(u, , 4, (2.10)

86 E. H. ROTHE
if
ec, = 8*5vuo> and if v - 21.
If (o*, $*) is another chart at y. it can be shown that
(2.11)
4+&o 9 4 N dF&u,; u). (2.12)
It follows from (2.10) and (2.11) and the definition of a tangent space that
dFe&o , u) induces a linear mapping t * = dF(x,; t), called the differential ofF at x0, of Ma0 into MzO .
Lemma 2.4 below follows immediately from the above definitions applied to the map F = 4-l of d(U) onto U C M, from (2.11) and from the non- singularity of d$.
LEMMA 2.4. u + t = d$-l(u,; u) is a map of E onto MnO . Morewer d@l(u,; u) = d+*-l(v,v) if and only zjc (2.11) holds.
We now assume that M is a Riemannian manifold. Then for each x E M the tangent space M, is a Hilbert space with a scalar product (s, t). . If then (U, $) is a chart at x0 and U, = +(x0) it follows from Lemma 2.4 that
($9 ho = W-Yu,; a>, d+-l(zl,; v)>, . (2.12)
with u, v uniquely determined by s, t. The right member is a positive definite symmetric form on E and defines therefore a scalar product (u, v> 4 on E depending on 4, For 4 = +, we write (u, v), for (u, v) 4 if (U, , q& is the family of charts at x0 indexed by the set I. Let u = u, , v = v, be the couple satisfying (2.12) for given s, t. Then by (2.12)
Cs9 t>% = C”$ 2 v*) $* = C”* 2 v*>j *
Let now x0 be a point of r. Then by (2.5) and Definition 2.2
(2.13)
where Ej2 is a hyperspace in E. Thus E = E,l + El2 for any space E,l spanned by an element e,l of E not in Ej2. We choose e,l as a unit vector orthogonal to Era in the metric given by the scalar product (,), defined in (2.13). We make the choice of e,l unique by the additional requirement that 4,‘([e,l) is exterior to V for small enough positive 5. (This choice is possible by Lemma 2.3.)
DEFINITION 2.3. The element n(xo) = {n,},E1 of MS0 where
*5 = 43Mxo); e5’)
is called the exterior normal to P at x0 .

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 87
The following theorem shows that a number of properties intuitively expected of an exterior normal actually hold for n(xJ.
THEOREM 2.1. (i) M=, is spanned by n(q) and pFO; (ii) n(x,) is orthogonal to p%-,,; (iii) for positive 5 small enough +;‘(cej’) is a pornt of M which is exterior to V while for all real 1, d$;‘(+(xJ; [ejl) is the point &(x0) of M,, .
For the proof we need the following
LEMMA 2.5. Let 0 be an open subset of E, and let E2 be a closed linear sub- space of E. Let F be a Cl map 0 ---f E, and let F2 be the restriction of F to O2 = 0 n E2. Finally let u2 be an element of 02. Then the restriction to E2 of the linear map E -+ E given by u + dF(u, , u) is the map E2 -+ E2 given by u + dF2(u2 , u).
The proof consists in a routine argument based on the definition of a differential. We therefore omit it and proceed to the proof of Theorem 2.1.
Proof of Theorem 2.1. (i) Let (U, , +j)iol be the charts for M at x,, indexed by I. Then a point of M,, is a set of points ui of E which are equi- valent under the relation
uk = d$k&‘(d(%>; %>- (2.14)
If (IV, , &) is the chart for p at x,, defined as in (2.6) (with N = #), then a point of Fe0 is a set of points vj which are equivalent under the equivalence relation
Ok = d’bk~;‘(~(%>; vj). (2.15)
We note that the “target space” for all dj is E, while the target space for #j is Ej2 since $i( IVj) = $j( Vi n 8) C Vj2 C Ej2.
Now &#;’ is the restriction of F = &+il from 0 = +j( Uj) to &(z+ f3 P). It therefore follows from Lemma 2.5 that d+k$;l(uo; v) is the restriction of d$k$;l(u,; U) from E to Ej2. Consequently, if (2.15) holds then (2.14) holds with Uj = ~j, uk = ~)lc, and we thus see that a point of pzjEg is also a point of M 5 , in other words pz;, C M, .
Let now t = {&, be an aibitrary point of Mz, . But Uj = Xej’ $- Uj2 with h real and uj2 E Ej2. Thus t = h{e,l} + {Uj”]. This proves assertion (i) of our theorem since {ejl} = n(xJ and {uj2} E pzO .
Proof of (ii). Let t be an arbitrary element of pzO . By Lemma 2.4 (with (b replaced by $j and E by Ej2) and again by Lemma 2.5 t is of the form
t = d~;%W,,); 4 = d~T’(drb>; 4 vj E Ej2.

88 E. H. ROTHE
Therefore by Definition 2.3 and by (2.13)
(4 4x0)>,, = <4SWo); 4, 43dj(xo>; ej’)>, = (vj , ejl)j .
But the scalar product at the right is zero since vi E Ei2, and since eji was chosen to be orthogonal to Ej2. This proves assertion (ii).
Finally the two assertions of (iii) follow immediately from Definition 2.3 and the paragraph preceding it.
We now recall the definition of the gradient of a Cr map f: MzO + R, the reals. Since R may be identified with its tangent space the differential df (x,,; t) is a real valued continuous linear functional on the Hilbcrt space ME, . Therefore there exists a unique element g(x,,) E Mz, such that
df(xo; 4 = <&oh 0% *
g((x,,) is called the gradient off at x,, . (In symbols Vf, or gradf). If f is Ck+l (k 3 1, theng is Ck. (For a proof see [8, p. 3131.)
The next lemma deals with relations between the gradient on M and the gradient on the linear space E.
LEMMA 2.6. Let (U, 4) be a chart at x0 E M. We set
h(u) = f#-l(u), y(u) = grad h(u), u E #( U) C E. (2.16) Let
uo = TYxoh t = d$5-yu,; u). (2.17) Then,
(Y(Uo), u> 4 = (g(xo), ozo * (2.18)
&o> = W(uo; Y(UO)? r(u3 = 4(x0; dxo)) (2.19)
II dxo)ll~o = II YWll? - (2.20)
Proof of (2.18). Using (2.17), the definitions involved and the chain rule we see that
<Y@oh a#a = 4 240; 24) = df#-yu,; 24) = df (p(uo); d-yu,; 24)) = df (x0; t) = (&Go), 0s *
Proof of(2.19). The two assertions of (2.19) are equivalent. We prove the first one. From (2.18), the definition of (,}, (given in the paragraph following (2.12)), and from (2.17) we see that
(‘@O>> t>zo = <r@oh u>r = @$w~o; Y(UoN3 4wJo; 4>zo
= w-‘(uo; Y(Uo), t>zo *
Since this equality holds for all t E M,, it implies the first part of (2.19).

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 89
Proof of (2.20). By (2.19) and the definition of (,),
LEMMA 2.7. Let (U, q5) be a chart for M. Let A be an open interval and let x be a Cl map A-++(U). F or 01 in A we adopt the usual dejinition x’(a) = dx(q 1). Moreover Zet
(2.21)
Using the notations of the preceding lemma we assert:
(9 If
then
rl’b) = -YhbN- (ii) If
then x’(4 = -&(~))lll g II29
rl’(4 = -rt~t4/ll Y v-
Proof. From (2.21) and the chain rule we see that
~‘(a) = d+(a); 1) = d+(x(ol); x’(a)).
Therefore if (2.22) holds then by (2.19) and (2.21)
--I’(“) = 4(44; g@(4) = Yww = Y(rl)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
which proves (2.23). If (2.24) holds we see from (2.26) (2.19) and (2.20) that
-~‘(4 = 4&+-4; &W/II g II2 = rC$HNll Y l12.
We now state our basic assumptions:
Let n/r be a connected Hilbert Riemannian C’ manifold (Y > 1) without boundary, let V be an open bounded subset of M, and f a real valued function defined on the closure P of V.
ASSUMPTION 2.1. M is a Fredholm manifold, and the boundary p of V is a hyper-submanifold of M (see Definition 2.2).

90 E. H. ROTHE
ASSUMPTION 2.2. (i) f is not constant in any ball; (ii) the gradient g off exists and is locally Lipschitz; (iii) if W is a subset of V on which /f 1 is bounded then /I g 1) is bounded on W.
ASSUMPTION 2.3. The Palais-Smale condition is satisfied, i.e., if f is bounded on a subset 5’ of v while /Ig 11 is not bounded away from zero on S then g vanishes in some point of closure S of S.
ASSUMPTION 2.4.
<g(x), wh! > 0 for every x E V. (2.27)
Here n(z) denotes the exterior unit normal to p at the point x E p (see Definition 2.3).
THEOREM 2.2. Let f be a real valued Cl function with domain v, and let Assumptions 2.1-2.4 be satisjed. Let x0 E V, and let x(a) be the gradient line through x,, , i.e., the solution of the differential equation
X’(4 = -g(x(4,
satisfying the initial condition
x(0) = xg .
(2.28)
(2.29)
Then x(a) E Vfor all nonnegative u for which x(a) is defined.
Proof. If the assertion were not true there would be an CQ > 0 such that
Xl = x(aJ E P, x(a) E v for 0 < OL < 01~ . (2.30)
Let now (U, +) be a chart for 44 at x, (of the type described in (2.3) to (2.6)), let T(U) be defined by (2.21), and let
u1 = &4 = Tw%)) = +%)- (2.31)
Let e1 be defined as in the paragraph preceding Definition 2.3 (with qGj = 4) and let y be as in Lemma 2.6. We then see from (2.17), (2.18) (with u = el), from the Definition 2.3 of the exterior normal, and from (2.27) that
<YW 6>4 = (g(4 n(xl)>,l > 0,
and, taking (2.28) and Lemma 2.7 into account, that
(7’W e’h < 0. (2.32)

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 91
We will now show that, in contradiction to (2.32),
(?‘W, oh5 2 0. (2.33)
and thus finish the proof of the theorem. We see from Lemma 2.3 that for positive and small enough 01~ - OL,
~(a) = --Xe’ + e2, h = X(a) > 0, e2 E E2. (2.34)
But it follows from (2.30) and (2.31) that ~(a~) E E2, and therefore from (2.34) that ~(a) - ~(a~) = --he’ + c2 with h > 0, and with .Z2 E E2. This proves that
+h4 - ~h>>/(~ - 4,e’> 4 > 0, (2.35)
since e’ is orthogonal to E2 (with respect to the scalar product (,)Q) and since a - 01~ is negative. (2.35) obviously implies (2.33).
DEFINITION 2.4. A point ‘ys E v is stationary forf if g(x,,) = 0. A number c is called a stationary value (or level) forfiff(y) = c for at least one stationary point y. The set of stationary points will be denoted by I’ and the set of stationary levels by il.
LEMMA 2.8. Let the set W be as in Assumption 2.2(iii) and assume it to be open. Then the set r n w is compact.
Since f is bounded on r n W the proof given for assertion (i) of Lemma 2.3 in [14] applies.
LEMMA 2.9. The set A is closed.
Proof. Let cr , cs ,... be a convergent sequence of stationary levels, and let
c0 = lim ci . i&xc (2.36)
We have to prove that C,EA. (2.37)
Let yi be a stationary point at level ci . Then on account of (2.36), f is bounded on the set {ri}. Since yi is an element of the open set V we can choose positive Li such that If(x) -f (ya)l < I 1 f or x E B(r, , &) C V. Then obviously f is bounded on the open set W = (Ji B(yi , &). Consequently, by Lemma 2.8, there exists a subsequence {m,} of the sequence {rj) which converges to a point y0 E r. Then we see from (2.36) that
This proves (2.37).

92 E. H. ROTHE
LEMMA 2.10. If the closed interval [a, b] contains no critical levels then there exists a positive m such that
II &)ll > m for x E f -l[a, b]. (2.38)
Proof. The lemma is an immediate consequence of Assumption 2.3 since f is bounded in the closed set f -l[a, b].
LEMMA 2.11. Let a, b and m be as in the preceding lemma. Let x(01, x0) be the gradient line through x,, , (as dejned in the Theorem 2.2). Then
x(01, x0> Efa if x0 EJb and a! 3 T = (b - a/m”).
Proof. If this were not true then f (T, x0) > a for some x0 E f -l[a, b], and for such x0 we would see from (2.38) that
a <f(xo) + loTg dt = f (xo) + l’ (g(x), $) dt
= f (x0) - lo’ 11 g(x)j12 dt < 6 - m2T = a.
LEMMA 2.12. Let r(c) denote the set of stationary points at level c (which may be the empty set). Let W be an open neighborhood of r(c). Then there exist real numbers a, b and T with a < c < b and T > 0 such that
x(T, xo) ~fa u W for XOEfb. (2.39)
Remark. We note that the existence of deformations having the property asserted for x(01, x0) in the lemma was proved by Palais [9a] and Browder [2] for Banach manifolds (without boundary) by the use of “pseudo gradients.” The proof below is given for completeness sake. It is divided into four steps.
Step 1. It is asserted that there exist a, , b, with a, < c < b, such that
I-O = I- n f -‘[a0 , b,] C W. (2449
We will show that there exists a C? > 0 such that (2.40) is true with a0 = c - d, bo=c+difO<d<J.
If this were not true there would exist sequence of positive numbers d, converging to zero, and a sequence of points yV with the properties
3/v ~f-‘[a - 4, c i- 41, yv E r - w. (2.41)
Since the sequencef(y,) is bounded we may by Lemma 2.8 assume that the yV converge. The limit y. is a stationary point which, by (2.41), lies on the

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 93
level c. Thus y,, E r(c) C W, and therefore yV E W form a certain v on. This contradicts the second part of (2.41).
Step 2. The set r” defined by (2.40) is compact, and the set v - W is closed. Therefore these two sets which by (2.40) are disjoint have a positive distance 5d, . Let now y. E To and let [(yo) be a number such that
0 < LXYO) < do 9 (2.42) I f(x) - f(Yo)l < 1 for x E Yro T 5(roNy (2.43)
g(x) satisfies a Lipschitz condition in B(y, , 35(yo)). (2.44)
(See Assumption 2.2). Since To is compact there exist a finite number of points y” (u
in To such that the balls B(y”, [(y”)) cover P. We set
Then
wi = u qY”,Jx(Y”)) forj = 1, 2,3. 0
(2.45)
PC WlC w2c w3c w.
We claim: there exists a positive T such that
4% x0) E w forO<cc<T and xosW2.
(2.46)
(2.47)
For the proof we note first that as a consequence of (2.44) g is bounded in W3, say
II &)I1 < M for x0 E W3. (2.48)
Let now 3co E W2. Then
II x - Y” II < Z(Y”)
for at least one cr. For such u we have obviously
(2.49)
qxo , 5(3p)) c B(Y”, 35(v)) c w3 c w. (2.50)
It follows from this inclusion in conjunction with (2.44) that g satisfies a Lipschitz condition in B(zco, ((y”)). Moreover (2.48) is satisfied for x E B(x,, c(v)). Therefore the local existence theorem for differential equa- tions allows us to conclude that X(CX, x0) is defined at least for j 01 / < i$-)/M, and that for such 01, x((Y, x0) E B(x, , c(y”)) C W2 (cf. 2.50)). This shows that the assertion (2.47) is satisfied with
T = ,=$n*., &Y”). (2.51)

94 E. H. ROTHE
Step 3. We assert the existence of a positive m’ such that
II &>ll > ml for x E Si =f-l[a, , b,] - W2. (2.52)
Indeed f is bounded on the closed set S. Therefore if the assertion were not true, Assumption 2.3, (2.40) and the inclusion (2.46) would imply the exist- ence of a point 7 E s’ n r C I’O C W2. This contradicts the fact that by definition the intersection S ~7 W2 is empty.
Step 4. Let T and m, be as in (2.51) and (2.52) resp. Let a, ,6, be as in Step 1. Finally let, a, b be a couple of numbers satisfying
a,<a<b<bo (2.53) and
b-a<Tqa. (2.54)
With this choice of a, b, and T we will show that (2.39) is satisfied. To this end we write j* as the union of three sets:
jb = ja u {f -l[u, b] n Wz} u {f -+z, b] n V - W2} (2.55)
If x0 E ja then (2.39) is obviously satisfied since f (x(oI, x0)) is non increasing in 01. If x0 is a point in the second summand at the right of (2.55) then, by (2.53), x0 E f -l[uo , b,] n m2, and our assertion (2.39) is satisfied by (2.47).
Finally let x0 be an element of the third summand in (2.55). Suppose first that
x(a, x0) c v - w2 for 0 < 01,< T. (2.56)
We then show that (2.39) holds by proving that
f (4T, 4) < a. (2.57)
If this inequality were not true we could conclude from the monotonicity of f (x(oI, x0) and from (2.53) that
bo 3 b > f (xo) > f+, ~0)) 2 f (CC ~0)) > = 2 a, for 0 < (Y < T.
Thus, by (2.56), x((Y, x0)) E 9 (cf. (2.52), and (2.52) holds with x = x((Y, x0) for these 01. Then
a -=c f(x(T, ~0)) = f (xo) + l= $ dT
= f (x0) - L* 1) g /I2 dr < b - q2T < a (cf. (2.54)).
This contradiction proves (2.39) if (2.56) is satisfied.

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 95
Suppose now that (2.56) is not true. Then since by assumption
x(0, x0) = x0 c v - w2,
there must be an cyl such that
0 < aI d T, 4% x0> I $V2 forO<ol<ol,, E IV2 for 01 = cir . (2.58) Let now x1 = x(o1r , x0) and let 5((B) = a(/?, x1) be the gradient line defined by dqdp = -g(q)), P(0, x1) = x1 . It then follows from (2.47) with x(01, x,,) replaced by ~((8, x1) that x@, x1) E W for 0 < fi < T. Therefore X(OL, x0) E W for 01~ < 01 < a1 + T since x(q + /3, x,,) = ~$3, x1). This proves that x( T, x,,) E W since 0~~ < T.
LEMMA 2.13. If the half open interval [a, c) contains no stationary values then J, is a deformation retract of fc .
Proof. For x0 E fC - Ja let x(~r, x0) be the solution of
dx - = - (f(x0) - a)g(x)/llg Hz, dol
x(0, x0) = x0 ,
and let
It is then easily seen that 6(a, x0) retracts fC ontofa (see, e.g., [14, Lemma 3.41). We conclude this section by proving the following theorem needed in
Section 4.
THEOREM 2.3. r is an ANR, i.e. an absolute neighborhood retract in the class of met&sable spaces.
The proof is based on the following lemma.
LEMMA 2.14. Let Y be an ANR. Let Yl and Yz be closed subspaces of Y whose union is Y and whose intersection is an ANR. Then Yl and Y2 are ANR’s.
This lemma is proved in [6, Proposition 9.1, p. 47 in conjunction with Theorems 3.1, p. 83 and 3.2, p. 841).
To apply this lemma to the proof of Theorem 2.4 we recall that every metric Banach manifold is an ANR (see [9, Corollary, p. 31). Thus Y = M and r are ANR. Setting Yr = r, Y2 = M - I/ we see that the lemma implies the theorem.

96 E. H. ROTHE
3. THE MORSE THEORY
Letfbe a real valued continuous function-defined on a topological space S.
DEFINITION 3.1. A real number c is called a critical value (or level) for f if for no two numbers a, b with a < c < b the setja can be deformed into the set ja . The set of critical values will be denoted by A,, .
ASSUMPTION 3.1. (a) A finite interval contains at most a finite number of cirtical values. (/3) If the half open interval [a, b) contains no critical values then fb can be deformed into ja .
For reference sake we state the following obvious consequence of Assump- tion 3.1 as a lemma;
LEMMA 3.1. If the closed interval [a, b] contains no critical values then jb can be deformed into ja .
If B r> A is a couple of subsets of S, and q a nonnegative integer then H&B, A) will denote the qth singular homology group of the couple (B, A). The coefficient group will always be supported to be a principal ideal ring.
LEMMA 3.2. If [a, b] contains no critical values then
HdJ, ,.fJ = 0 for all 4. (3.1) This is an immediate consequence of Lemma 3.1 and well known properties
of the homology groups.
LEMMA 3.3. If for all intervals [a, b] containing the real number r the homo- logy group %(.h , fJ is dff i eren t f rom zero for some q then r is a critical value.
Proof. If r is not critical then by Assumption 3.1(a) there exists an inter- val [a, b] containing r and no critical values. For such interval (3.1) holds by Lemma 3.2 for all q. This obviously proves the lemma.
LEMMA 3.4. Let c be a critical value. Then H,,( jb , jJ does not change as long as c is the only critical value in [a, b].
Proof. We have to show that
H&ib da) = H,(h ,L), (3.2)
if the intervals [a, b] and [01, fi] both satisfy the condition of the lemma. It is easy to see that we may assume b > /3 > c > 01 3 a. We will first show that
&(h 3 jaa) = H,(jb 2 .fx,,. (3.3)

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 97
fu’ow the interval [a, a] contains no critical value. This fact implies by Lemma 3.2 that
f&(3, ,3a> = 0. (3.4)
It also implies that fa is empty if ja is empty since, by Lemma 3.1, fa can be deformed into fa . Therefore in this special case the assertion (3.3) reduces to fU3tJ = HAJtJ. But if Ja . is not empty then it is well known that (3.4) implies (3.3) (see [5, 1.8.11).
This proves (3.3), and the isomorphism
&(3b 932 * KAJ, ,3J,
is proved in a similar way. But (3.3) and (3.5) imply (3.2). Lemma 3.4 allows us to make the following definition.
(3.5)
DEFINITION 3.2. The qth critical group C,(c) at the critical level c off is defined by C,(c) = H,( fb , jr,) w h ere a < c < b and where c is the only critical level in [a, 61.
Since the coefficient group G of the homology theory is supposed to be a principal ideal ring. The classical decomposition theorems hold if G is finitely generated. If G is not necessarily finitely generated we have the following definition.
DEFINITION 3.3. Let T be the torsion submodule of G. Then the rank c(G) of G is defined as follows: if the quotient module G/T is not finitely generated then c(g) = CO; if G/T is finitely generated then t(G) = {(G/T), i.e., the number of elements in a base of the (free) group G/T.
LEMMA 3.5. If c(G) is jfinite then we have the direct decomposition
G=F/T, (3.6)
where T is as above and where F is a free module. Moreover
5(G) = 5(F). (3.7)
The proof follows easily from a well-known lemma (see, e.g., [5, p. 133, Lemma 6.31).
We now introduce notations which will be used in Theorem 3.1 below. Let a < b be two number which are not critical values and let cr < c2 < . . . c, be the critical values in [a, b]. Moreover let a, , a, ,..., a,,, be numbers such that

98 E. H. ROTHE
In addition we set
A =3a, B =3t,, 4 =3aa, 9
Caa = C&a) = K&% 9 4x-A
M,” = S(Cqa), M, = 5 M,“, a=1
R&s b) = W,(B, A)), a = 1, 2 ,,,., N.
(3.9)
(3.10)
THEOREM 3.1. For the validity of the inequality
(3.11)
each of the following three conditions is sujjicient
(i) [(&(A, , A,) < co, a~ = 1, 2 ,..., N.
(ii) the critical groups Cga are Jinitely generated,
(iii) the coeficient group G is a jield.
Moreover if (i’) is condition (i) with the additional proviso that
(iv) M,” = [(Cam) < co, 01 = l,..., N
then each of the conditions (i’) and (ii) is suficient for the validity of the inequality
$O(-l)q-siM,(a,b) >~$O(-l)‘+sR,(a~b). (3.12)
Remark. If f is bounded then for small enough a and large enough b, %(a, b) is the qth Betti number of S and the “Morse numbers” M, are independent of a and b. Thus inequalities (3.11) and (3.12) are in this case the classical Morse inequalities.
Proof of Theorem 3.1. If one of the numbers Mpl,..., MaN is in&rite then M, = co by (3.10), and (3.11) is trivially satisfied. Therefore for the proof of this inequality we may make the additional assumption (IV), and thus replace (i) by (i’). But under the latter condition the proof of (3.11) and (3.12) inequalities is the same as the one given by Pitcher for the finite dimensional case ([ 11, Section 1 I]) and is therefore omitted.
To give the proof under condition (ii) it will now be sufficient to prove that this condition implies (i’). Since (iv) is obviously implied by (ii) the proof will be finished if we verify that (ii) implies (i) by showing that the groups &(A,, A,,) are finitely generated. This is done by induction in

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 99
a: H&4, , A,) is certainly finitely generated since this group is C,l. We assume that HJA,-r , A,) is finitely generated and consider the part
of the homology sequence for the triple A, CA,-, CA, . Here the two extreme groups are finitely generated, the one at the left by induction assumption, the one at the right because it is the critical group Cpa. From this and the exactness of the sequence it follows that the kernel K of the map i* as well as the group H,(& , &J/K are finitely generated. This obviously implies that Ti,(A, , A,) is finitely generated. This completes the proof of the theorem since (iii) and (iv) together imply (ii).
We now return to the situation of Section 2 by setting S = r.
THEOREM 3.2. Theorem 3.1 is valid if S = r and ;f Assumption 3.1 is replaced by Assumptions 2.1-2.4 and the additional
ASSUMPTOIN 3.2. A finite interval contains at most a finite number of stationary values (cf. Definition 2.4).
In fact Assumptions 2.1-2.4 and 3.2 imply Assumption 3.1 as the following two lemma show.
LEMMA 3.6. A critical value is a stationary value, i.e., A, CA.
Proof. We show: if c is not a stationary value then c is not critical. Indeed by Assumption 3.2 there correspond to a nonstationary c two numbers a, b with a < c < b such that [a, b] contains no stationary values. By Lemma 2.11 the set j,, can be deformed into the set ja but this implies that c is not critical (cf. Definition 3.1).
LEMMA 3.7. The set A, of critical values satisfies Assumption 3.1.
Proof. Assumption 3.2 together with Lemma 3.6 show that the (a) part of Assumption 3.1 is satisfied. The (/3) part follows immediately from Lemma 2.13 if the interval [a, b) which by assumption is free of critical levels is also free of stationary levels. Suppose now [a, b) contains stationary values. By Assumption 3.2 there are only a finite number, say sr > s2 > ... > s, . Since the si are not critical values there exist ai , bi such that ai < si < b, and such that jbbi can be deformed into jai . Obviously we can choose the ai and bi in such a way that in addition
b > b, > s, > a7 > b,-, > s,-~ > a,-, > ‘.. > b, > s1 > a, > a

100 E. H. ROTHE
(with an obvious modification if a is a stationary value). Now there are no stationary values in [b, , b). Therefore fb can be deformed into J,, (again by Lemma 2.13). ButX, can be deformed intofall . Going on this way we obtain deformations whose product deforms fb into ja .
LEMMA 3.8. C,(c) M H,(fa ,fJ if c is the only critical oalue in [c, b].
This is an immediate consequence of the preceding lemma and Definition 3.2 together with the deformation invariance of the homology groups.
Remark. In Theorems 3.1 and 3.2 only critical levels are considered. But if we were to define critical groups C,(c) (in analogy to Definition 3.2) for stationary but not critical levels c then it is easily seen from Definition 3.1 that these C,(c) are zero groups. Therefore M,(c) = [(C,(c)) = 0. Thus there would be no change in the inequalities (3.11) and (3.12) if all stationary levels were taken into account, i.e., if A,, is replaced by A.
We now consider critical points.
DEFINITION 3.4. The point y0 E Y is called a critical point off is for no neighborhood W of ‘ye the set f,, n W u {ys}, can be deformed into the set foe n W where c,, = f (‘y,,). The critical point y,, is called isolated if there exists a neighborhood of y,, containing no other critical point.
LEMMA 3.9. If y,, is critical then y,, is stationary. Proof. Suppose y,, is not stationary and let co = f (3/s). Then g(y,,) # 8.
Therefore there exists a neighborhood W of y,, in which /I g(x)// is bounded from below by m = 11 g(rJ1/2. 0 n account of this fact it is easy to construct a deformation deforming fc, n W u {~a} into fc, n W by using the gradientline through ‘y,, (cf. (2.22)).
LEMMA 3.10. Let W and W, be open neighborhoods of the isolated critical point y,, . Suppose that W and WI contain no other critical point. Then, with
co = f (‘yoh
Proof. We may assume that WI WI (otherwise consider WI n W). Then the lemma follows by excising the set U = (W - WI) n fc, from the couple at the right member.
This lemma allows us to state the following.
DEFINITION 3.5. The qth critical group C,(y,) of the isolated critical point y. is defined by
Cho> = f&(f,, n W u ~~~Lfc, n Wh

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 101
where W is an open neighborhood of y,, containing no other critical point and where ca =f(yJ.
A remark analogous to the one following Lemma 3.8 can be made con- cerning the definition made concerning the definition of groups attached to an isolated stationary but not critical point.
THEOREM 3.3. Let the assumptions of Theorem 3.2 be satisjed. Suppose that c is a critical level at which there are only a finite number of critical points, say
Yl , yz ,..., yr . Let b > c be such that the interval (c, b] contains no critical values. Then
C*(c) 7% f&(fc ,fc) r=x k CAYi), (3.13) i=l
where .Z denotes the direct sum.
For the proof we need Lemmas 3.11 and 3.12.
LEMMA 3.11. Let the assumptions of Theorem 3.3 be satis$ed. Let x(01, x0) satisfy
dx - = -(f hi) - 4 dx)lll &)l12~ da:
Then
40, x0) = x0 , x0 ef-l(c, b]. (3.14)
(i) df (~) - = -(f@o) -4 da (ii) c<f(x(cd,Xo)<bfoY O<a<l,
(iii) &nf(x(q x0) = C,
(iv) j$ x(01, x0) exists.
For assertions (i)-(iii) the assumption that the critical set at level c is finite is not necessary.
Proof. The elementary proof of assertions (i)-(iii) may be found in [12, Lemma 5.31. We turn to the proof of (iv) which is an modification suited to the present situation of the proof given in [12, Theorem 5.1)].
Let {(a) be the distance of the point x(01, x0) to the critical set r(c) at level c, and let
We distinguish two cases.

102 E. H. ROTHE
Case I. [ > 0. In th’s 1 case there exists a positive constant m such that
II &)ll > m forxES={z=X(a,&)O<a<l1). (3.15)
Indeed otherwise we would from Assumption 2.3 and the boundedness off on S (quaranteed by (ii)) conclude the existence of a stationary point y0 in the closure of S. There would then be a sequence 01~ , 01s ,,.. such that
pi X” = Yo if x, = x(s) x0). (3.16)
Now because of our assumption % > 0 the point y. cannot belong to I’(c). Thus y. E r - r(c). But this is also impossible. For by Assumption 3.2 there exists a d > 0 such that
r - r(c) C&--d U {f 2 b + 4. (3.17)
But it follows from (3.16) and assertion (ii) above that c < f (yo) < b. Thus the existence of an m satisfying (3.15) is established. From this in
conjunction with (3.14) we see that for 0 < a’ < a” < 1
II+“, x0) - x(a’, *0)ll = I[ C” x’ da (j < 1 f (x0) - c)l (a” - a’)/m
< (b - c) (a” - a’)/m.
By Cauchy’s principle this implies the validity of assertion (iv).
Case II. r = 0. Then there exists a convergent sequence {ai} with
O<CL,<l and lim ai = as < 1 i-m
such that the sequence ~(a~ , x0) converges to one of the points yr , ys ,..., yr , say to yi:
& x(ai ,x) = n . (3.18)
We must have
~=limai=l. i+w (3.19)
For otherwise 0 < a,, < 1, and by assertion (ii) above,
f&f @(ai , x0)) = f @(a0 2 x0) 3 c.
But by (3.18) this limit equals f (yl) = c.

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 103
Thus (3.19) is true, and we may assume that the 01~ are monotone increasing. We now prove assertion (iv) by showing that
lim x(01, x) = y1 . e-1-
(3.20)
To do this we first exhibit a positive [,, of the following property: if 0 < i$ < 5, then there exists a positive constant m = m(& , &,) such that
il &)li > m for x E 45 , 5,) - Ql ,5d (3.21)
We choose for {,, a positive number satisfying
Ll < ,=y&,, II Yi - Yl ll/2, and I f(x) - fbd < mW/2, b - 4
for x E B(rl , [,,) where d is as in (3.17). Since then c - d/2 <f(x) < b for x E B(y, , to) it is (cf. (3.17)) easily verified that the closure of the set appearing in (3.21) contains no point of r = r(c) u (I’ - r(c)). The existence of an m of the asserted property follows then from Assumption 2.3.
To prove the assertion (3.20) we now make the assumption that it is false. Then there exists a positive 8 and a monotone increasing sequence {o+‘} such that
II x(9’, %), Yl II t 6 (3.22) and
lim oLi’ = 1. j+m (3.23)
Let now j3 be a positive number such that g(x) is Lipschitz in B(y, , 5/?) (cf. Assumption 2.2) and such that
0 < 5/3 < min(6, Co). (3.24)
Then (3.21) is satisfied if we choose
51 = P. (3.25)
Now by (3.18) there exists an integer n,, such that
4% 3 43) E Wl > 8) for 1z > n, , (3.26)
On the other hand, by (3.23), (3.22) and (3.24) there corresponds to each n > n,, an integer 11’ = n’(n) such that
and
ano < a, < a;* < 1 (3.27)
II x(4&* > x0), Yl II b v. (3.28)

104 E. H. ROTHE
The proof of (3.20) will be finished by showing that for some 71’ = n’(n) with n > n,
x(4? , x0> E ml 3 4P)Y (3.29)
in contradiction to (3.28). To this end we note first that the relations (3.26)- (3.28) imply the existence of an a,* for which
0 < anO < an < a:,* < a;, < 1
and
4an*, x0> E %l t 3P)
where B denotes the boundary of B. Then by (3.30) and (3.19)
(3.30)
(3.31)
lima,* = 1. n+m (3.32)
We now consider the ball B, = B(x(cx,*, x0), ,9). Obviously
It therefore follows from our first assumptions on fi and from (3.21) that the right member of (3.14) is Lipschitz in B, and there bounded by a constant independent of n. If M is such a constant it follows from the definition of ~(a, x,,) as solution of the differential equation (3.14) that
II 4% x0), x(%*9 %)ll < fJff I a - an* I > (3.33) if
I a - an *I <BIM. (3.34)
We now choose a fixed n > n,, such that 0 < 1 - a,,* </3/M. This choice is possible by (3.32) and (3.30). It then follows from (3.30) that (3.34), and therefore (3.33), is satisfied with (Y = aI, . Thus ~(a:, , x,,) E B, which obviously implies (3.29).
LEMMA 3.12. Under the assumptions of Lemma 3.11 the set J, is a deforma- tion retract of Jo .
Proof. Let ~(a, x,,) be as in Lemma 3.11 and let
qx, , a) =
1
X(% x0>, if x0 9 if XcJE&-ffc, O<ff<l,
lp(% x0>, if XO~X -fc 9 a = 1,
x0 2 if x0 EfC ? O<a<l.
Qil 9 a) obviously retracts J,, onto jC, (for the continuity of 6(x, , a) cf. the appendix in [14]).

THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 105
We now turn to the proof of Theorem 3.3. From Lemma 3.12 and Defini- tion 3.2 we see that C,(c) m H,,(jc , ja) f i a is such that c is the only critical level in [a, b]. From this the first part of (3.13) follows sincef, can be deformed into Ja (cf. Lemma 3.7).
Now the set fc can be deformed into the set fc u r(c) by the deformation given by the solution of (2.22). Therefore
For i = 1, 2,..., r let now Bi be a ball with center yi in whose closure yi is the only critical point, (for the proof of the existence of such a ball cf. the argument for the existence of I&, in the paragraph following (3.21)). If we let W = (Ji Bi and excise the set fc U r(c) - W from the couple at the right of (3.35) we see from (3.35) and the excision theorem [5, VIII, 9.11 that
C,(c) - &(fc n Wu W,fc n W).
But the group at the right is isomorphic to the direct sum of the groups H,(fc n Bi u yi , fc n Bi) as is seen from the addition theorem [5, I, 13.21 and the definition of W. By Definition 3.4 this proves the second part of assertion (3.13).
We now turn to a discussion of the Morse inequalities (3.12) in the case that all critical points are isolated.
THEOREM 3.4. Using the notations introduced in the paragraph immediately preceding Theorem 3.1 we suppose that the critical set r(c,) at level c, consists of a finite number of points ‘yai (i = 1, 2 ,..., ror , 01 = 1, 2 ,..., N). Let mcri denote the rank of C’,(Y,~) (see Definition (3.5)). W e su pp ose moreover that the groups C,(y,“) are finitely generated. Then the Morse inequalities (3.12) hold with
M, = g 2 m,i. a=li=l
Proof. The theorem is an immediate consequence of Theorems 3.2 and 3.3.
Remark. Suppose all ymi are non degenerate. (For the definition of non degeneracy and of the index of a nondegenerate critical point see, e.g., [8, p. 3071). Then the critical group CQ(yoli) is isomorphic to the coefficient group if q equals the index of yori, and 0 otherwise as proved in [8, p. 3361 (for a different proof see [15; Theorem 2.1 and Corollary to Theorem 2.21). It follows that the conclusion of the preceding theorem is valid in this case. It follows moreover that M, equals the number of critical points of index q.

106 E. H. ROTHE
Cf. [8, p. 3381 where the Morse relations are proved in the case of non- degeneracy if the manifold is without boundary and the coefficient group is a field.
For another case in which the CP(rai) are finitely generated see [14, Theo- rem 7.31.
4. A LUSTRRNIK-SCHNIRELMAN THEOREM
We first recall some basic definitions. Let A be a subset of the topological space X. Then cat (A, X), the category of A with respect to X is defined as follows: cat(A, X) = 1 if A in contractible on X to a point of X; cat(A, X) = K if k is the smallest integer such that A can be covered by K closed sets each of which is of category 1 with respect to X; if no such k exists then cat(A, X) = co.
For positive integer K < cat(X, X) and real valued f with domain X the Lusternik-Schnirelman number m, = mk(f, X) is defined as follows: let S, be the family of those subsets A of X for which cat(A, X) > k. Then
THEOREM 4.1. Let f and V satisfy the Assumptions 2.1-2.4. In addition f is supposed to be bounded below. Then
(i) each jlnite m, is a stationary value off,
(ii) the number of stationary points is not smaller that cat(X, X),
(iii) ifm, = m,,, = *.. mktn and mk isfinite then cat(r(m,), X) > n + 1. Here r(m,) denotes the set of stationary points at level mk .
Proof. The following facts were proved earlier or follow directly from the definitions involved; F is a metrizable absolute neighborhood retract (Theorem 2.3). The intersection of the set of stationary points with f -l[a, b] where [a, b] is a finite interval is compact (Lemma 2.8). Each point of v has a neighborhood contractible to that point (Corollary to Lemma 2.3). But these facts together with Lemmas 2.11 and 2.12 are known to ensure the validity of our assertion (See [2, Theorems 2 and 31.)
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THE THEORIES OF MORSE AND LUSTERNIK-SCHNIRELMAN 107
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