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An F-space sampler. (English) Zbl 0556.46002
London Mathematical Society Lecture Note Series, 89. Cambridge etc.: Cambridge University Press. XII, 240 p. £15.00; \$ 24.50 (1984).
In this interesting book, the authors collect some important aspects of the modern theory of F-spaces, i.e. complete metric topological vector spaces, and, in contrast to most books on functional analysis, the text concentrates on the study of the non-locally convex (non-l.c.) spaces in this class. The classical examples are $$\ell_ p$$, $$L_ p$$ (on [0,1]), and the Hardy spaces $$H^ p$$ on the unit disk, for $$0<p<1$$, as well as $$L_ 0(\mu)$$, the space of all (equivalence classes of) measurable functions on the finite measure space ($$\Omega$$,$$\Sigma$$,$$\mu)$$, equipped with the F-norm $\| f\|_ 0=\int \quad \frac{| f(x)|}{1+| f(x)|}\quad d\mu (x)$ (inducing the topology of convergence in measure). After some preliminaries, the book begins with a discussion of these concrete examples before going on to general results and important counterexamples. Throughout the text, the classical spaces play a prominent role, and some parts of the book develop their structure theory as compared with the theory of the corresponding Banach spaces $$\ell_ p$$, $$L_ p$$, and $$H^ p$$ for $$1\leq p\leq \infty.$$
In the case of non-l.c. F-spaces X, duality theory breaks down due to the lack of the Hahn-Banach theorem, and a number of new phenomena appear which the authors study in detail: There may be proper closed weakly dense (PCWD) subspaces of X, and, in fact, whenever X has trivial dual or whenever X is a non-l.c. separable F-space with separating dual, then X always contains a PCWD subspace. Also, an F-space with Hahn-Banach extension property (HBEP) must be locally convex. Next, there even exist rigid spaces X, i.e. quasi-Banach spaces whose algebra of (continuous linear) endomorphisms reduces simply to multiples of the identity operator. (Actually, there is a continuum of non-isomorphic rigid subspaces of $$L_ p$$ when $$0\leq p<1.)$$ On the other hand, an F-space X may well have trivial dual, but admit non-zero compact endomorphisms. The important idea of transitivity is related to these results: An F-space X is said to be transitive if, for x,y$$\in X$$ with $$x\neq 0$$, there always exists an endomorphism T of X with $$Tx=y$$. The spaces $$L_ p$$ $$(0<p<1)$$ are transitive, and for transitive quasi-Banach spaces with trivial dual there are no non-zero strictly singular (and hence no non-zero compact) endomorphisms. Finally, $$L_ p$$ $$(0\leq p<1)$$ contains compact convex sets with no extreme points (so that the Krein-Milman theorem fails badly).
There are a number of other interesting topics which the authors treat, among them three space problems and the notion of K-space as well as its relation to lifting theorems for operators: An F-space X is a K-space if, whenever Y is an F-space and L is a subspace of Y with dimension one such that Y/L is isomorphic to X, then L is complemented in Y. X is a K-space if and only if, whenever Y is an F-space and $$E\subset Y$$ is a finite dimensional subspace of Y, then any (continuous linear) operator $$T:X\to Y/E$$ can be lifted to an operator $$\tilde T:X\to Y.$$ E.g., $$\ell_ p$$ and $$L_ p$$ are K-spaces provided $$0<p<\infty$$ and $$p\neq 1$$; also, $$L_ 0$$ is a K-space, but $$\ell_ 1$$ is not!
Since the book contains a wealth of material, it is impossible to discuss all of it here. Many of the results are due to the authors, and most are (closely) related to their own work, but we should also mention the names of Klee, Duren-Romberg-Shields, Stiles, J. H. Shapiro, Pallaschke, Waelbroeck, Turpin, s. Dierolf, Ribe and Drewnowski (plus some of the people known from Banach space theory such as Pełczyński, Kwapień and Pisier).
Table of contents (including some remarks on material not yet mentioned in the first part of this review). Chapter 1: Preliminaires (F-norms; locally bounded and p-convex spaces; closed graph and open mapping theorems; bases). Chapter 2: Some of the classical results $$(L_ p$$, $$\ell_ p$$; Banach envelope; Orlicz function and sequence spaces). Chapter 3: Hardy spaces (linear topological properties; the Banach envelope and PCWD subspaces of $$H^ p$$- this includes Aleksandrov’s theorem $$L_ p(T)=H^ p+\bar H^ p$$, $$0<p<1$$, and a discussion of PCWD-subspaces satisfying certain invariance properties). Chapter 4: Hahn-Banach extension property (M-basic sequences; minimal and atomic spaces; example of a non-l.c. quasi normed F-space with reflexive Banach envelope and a proof of the fact that the Banach envelope of a non-l.c. locally bounded F-space cannot be B-convex). Chapter 5: Three space problems (quasi-linear maps and K-spaces; the Ribe space; the type of quasi-Banach spaces). Chapter 6: Lifting theorems (for $$L_ 0$$ and $$L_ p$$, $$0<p<1$$; connection with K-spaces). Chapter 7: Transitive spaces and small operators (commutative quasi-Banach algebras; semi-Fredholm operators and strictly singular endomorphisms; compact operators on spaces with trivial dual; example of a non-l.c. ”pseudo-reflexive” space; a proof that $$L_ p$$ cannot be the domain of a non-zero strictly singular operator for $$0<p<1)$$. Chapter 8: Operators between $$L_ p$$- spaces, $$0\leq p<1$$ (characterization of endomorphisms of $$L_ 0$$ and $$L_ p$$, $$0<p<1$$, by series representations and some consequences, e.g. every complemented subspace of $$L_ 0$$ is isomorphic to $$L_ 0)$$. Chapter 9: Compact convex sets with no extreme points (needle point spaces contain compact convex sets with no extreme points; $$L_ p$$, $$0\leq p<1$$, is a needle point space; some open questions, e.g. concerning the fixed point and simplicial approximation properties). Chapter 10: Notes on other directions of research (1. Vector measures, 2. Operators on spaces of continuous functions, 3. Tensor products, 4. The approximation problem, 5. Algebras, 6. Galbs).
The list in Chapter 10. (quote) ”is by no means intended to be complete”. E.g., the authors do not mention the interesting work of Etter, Gramsch, Waelbroeck et al. on holomorphic functions with values in non-l.c. spaces and its applications to non-l.c. topological algebras. Moreover, quasi- Banach spaces and non-l.c. F-spaces occur in a very natural way when dealing with operator ideals on Banach (or even Hilbert) spaces, cf. Pietsch’s book, but this is not included in the list. However, the authors apologize: ”Some readers will feel that certain topics should have been (or sould not have been!) covered - obviously the selection of material in a volume of limited length has to be somewhat personal.” Also, they say: ”Although these notes have three distinct authors, we have tried to be as consistent as possible. There are certain inconsistencies in style and notation which we hope will not unduly distract the reader.”
But, in general, the authors succeed very well in their aim to ”present some aspects of the theoy of F-spaces which we hope the reader will find attractive”. Indeed, they demonstrate that ”with the aid of fresh techniques one can develop a rich and fulfilling theory” of non-l.c. F- spaces. The book is reasonably self-contained, a good source for research mathematicians and graduate students in functional analysis and a welcome addition to the literature.
Reviewer: K.-D.Bierstedt

##### MSC:
 46A04 Locally convex Fréchet spaces and (DF)-spaces 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46B25 Classical Banach spaces in the general theory 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness) 46H05 General theory of topological algebras 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46A35 Summability and bases in topological vector spaces 46A45 Sequence spaces (including Köthe sequence spaces) 46A55 Convex sets in topological linear spaces; Choquet theory 46G10 Vector-valued measures and integration 46M05 Tensor products in functional analysis 47A53 (Semi-) Fredholm operators; index theories 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 52A07 Convex sets in topological vector spaces (aspects of convex geometry) 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)