If Y is a finite-dimensional linear subspace of the normed linear space X, then Y is a closed subspace of X. If K is a compact subset of the normed linear space X, then there is a sequence in X such that 0 and K is contained in the closed convex hull of { x. Compact subsets of C(K) spaces for compact metric K Let (K, d) be any compact metric space, denote by C(K) the Banach space of continuous scalar value.
If there is a set in the normalized linear space for which weak is 0, then there is a set of convex combinations of the x such that lime II series are members of A. Let F be a finite-dimensional linear subspace of the infinite-dimensional Banach space X, and let e > 0.
Suppose there is a bounded linear projection P : X - X onto the closed linear span [zn I of the z,. P (it is not difficult to see), the sequence (bk/Ilbklp is a basic sequence equivalent to the unit vector basis (ek) of 1P, and the closed linear span of the bk is augmented in. This is an easy follows from the Eberlein-Smulian theorem that a Banach space is reflexive if and only if each of its separable closed linear subspaces is.
We denote by ca(E) the linear subspace ba(E), which consists of countably additive measures on E.
A(C\D)+A(D\C) - A(COD) S 8,
IIg-II
Our calculations alert us to the proximity of g-, in normalization, to hn, in normalization. Now for the proof of the main theorem we start with a non-weak compact closed unit sphere Bx of X. 0 One surprising use of the Dieudonne-Grothendieck criterion in tandem with Phillips' lemma is to study.
This is an easy consequence of the Stone representation theorem and the Stone-Weierstrass theorem. The domain of the map is dense, and so is its range (thanks to K's total disconnection and the Stone-Weierstrass theorem). Let K be as described in the paragraph preceding the statement of the theorem and suppose it is weak* zero in C(K).
We provide full evidence of the relevant facts only in case 15 p 5 2; what happens (and why) in case p > 2 is explained in the exercises. Our presentation of many of the results of this chapter was inspired by an unpublished manuscript of J. Vitali (1907) showed that if a series of Lebesgue integrable functions on [0,1] converges almost everywhere to f, then fo f (s) ds and lim fo f,(s) ds exist and are equal if and only if the indefinite integral of f is uniformly absolutely continuous with respect to Lebesgue measure.
The weak sequential completeness of ca(E) and L1(X) is an easy consequence of the Vitali-Hahn-Saks and Nikodym convergence theorems. We cannot leave our discussion of the Banach-Saks-Szlenk theorem without recalling the now famous discovery of J. The uncovering of the sum operator as a universally non-weakly compact operator was the work of J.
Tzafriri and serves as an excellent example that clarity of view improves over the years. On the way to establishing equality, take any vector of the form E =inb,x and look at E' tbix, r= [x1,. The proof of this statement (and consequently Theorem 3) will be induction on the number I of nonzero terms in vectors of the form E-laixi.
Indeed, McShane gave a proof of the uniform convexity of L, (µ, X), which to include the vector case considerably simplified existing proofs for plain old L"(µ). Notes and Remarks 141 behavior asymptotic of the convexity modulus of LP(µ); exactly, for any non-trivial measure fL,.
Figiel and G. Pisier (1974) gave much more in response to
- h(x)=F(h)- JhdtsJhdsS Jadµfor all aGA(K)such that hsa
For a summary of these events, the reader is referred to the monograph by W. 1979) and relevant parts of Lindenstrauss-Tzafriri's books. Suppose that x is an extreme point of K and p is a regular Borel probability measure on K that represents x. Rainwater's theorem gives a strong hint about the control extreme points of the double-ball exercise under weak convergence.
Let C, Co and Cl be closed bounded convex subsets of the Banach space X and let e > 0. As in the proof of the Superlemma, we see that the diameter of K \ Dr is strictly less than. Then the set of points with weak *-norm continuity of the identity map idx of K meets the set ext K of extreme points of K in a set that is a dense 9a subset of (ext K, weak.
To prove the present result, we will follow the direction of the proof of Theorem 7. Further, we saw in the proof of Choquet's theorem that in a metrizable compact convex set, the complement of the extreme point set is a countable union of sets of closed. ; the set of extreme points is a 9Fa. Take special note of the inclusion of 9(S) within Bcs(s)., making it natural to consider 9(S) in its weak* topology.
Point 3 is closer to the spirit of the theorem itself. Let S be a norm-separable subset of Bx. Then norm Borel subsets of S and sets of the form S n B, where B is a weak* Borel subset of B. Let °d/ denote the collection of all subsets of S that have the form S n B, where B a is a weak* Borel subset of Br. amp;' contains a base for the norm topology S, namely the sets S n B, where B is the closed sphere X*. Our next lemma, written by John Elton, points out serious limitations on the resolution of the set of extreme points.
First, we must pay particular attention to the following: if (u;) is a normalized block basis built on (xn ), then (u.,) is weakly zero. In view of the previous section, the closed linear span of (xn) entire X* is P*x* = x*Pm. Of course, is equivalent to the unit vector basis of 1i, and the closed linear span U of un is itself an isomorphic copy of 1i.
Our treatment of the simplest elements of the barycentric calculus is clearly indebted to R. Since the publication of the above-mentioned monographs, some surprising developments in the theory of integral representations have appeared, closely related to the material of the section entitled Elton's Theorem. The dentahility theorem (as it has come to be known) was finalized in the summer of 1973 and heralded the beginning of a period of excitement in the geometric affairs of the Radon-Nikodym property.
Some properties of the set of extreme points of the unit ball of a Banach space. 0 The proof of the previous lemma, viewed from the right perspective, highlights the special role of the Gaussian distribution in the current setting. We can now state and prove the "fundamental theorem of the metric theory of tensor products".
Grothendieck's Inequality and the GLP Idea Cycle 179 Let us rewrite the relevant parts of the above identity:. Note that each of the terms on the right involves quantities related in some way to Kai j)Ir. This result, due to Lindenstrauss and Zippin, is one of the real treasures in the theory of Banach spaces.
The importance of the Lindenstrauss-Pelczynski paper for the revival of Banach space theory cannot be overstated. Interestingly enough, many of the other proofs of Grothendieck's inequality have come in applications of Banach space ideas to other areas of analysis. As usual in such matters, the exact determination of the best constant that works in Grothendieck's inequality has aroused considerable curiosity.
Krivine has provided a scheme that suggests the best value of Grothendieck's constant and probably sheds a lot of light (for those who will see) on the precise nature of Grothendieck's inequality. The development of the structure of 2p spaces was one of the crowning successes of Banach space theory after the Lindenstrauss-Pelczynski breakthrough. Pisier (1978) showed that L1[0,1]/R has the Grothendieck property if R is a reflexive subspace of L1[0,1]; therefore, 21-spaces are not alone in enjoying the Grothendieck property.
