Inversion formulae for the probability measures on Banach spaces

Geometric duality theory of cones in dual pairs of vector spaces

For a double array of independent random elements {V mn ,m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ i=1 m Σ j=1 n V ij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

Let B be a real separable Banach space and R: i'->ia covariance operator. All representations of R in the form 2en ® e, (e, n > 1} c fi, are characterized. Necessary and sufficient conditions for R to be compact are ob- tained, including a generalization of Mercer's theorem. An application to character- istic functions is given. 1. Introduction. The study of covariance operators is a major component in the theory of probability measures on Banach spaces (10), (9), (1). The covariance operator of a strong second-order measure is always compact (2); however, the covariance operator of a weak second-order measure need not be compact. In this paper we first characterize series representations of covariance operators, and then give a set of necessary and sufficient conditions for a covariance operator to be compact. The classical Mercer's theorem (7) can be obtained as an immediate corollary. These results are then applied to extend a result of Prohorov and Sazanov (6) on relative compactness of probability measures from Hubert space to Banach space. 2. Definitions and notation. B is a real separable Banach space with norm || ■ || and topological dual B*. A linear operator R: B* -» B is a covariance operator if 7? is symmetric and nonnegative: {Ru, u> = and 0, for all u, v in B*. A probability measure ii on the Borel a-field of B is said to be weak second-order if fB(x, «>2 dii(x) > = j = I {x — m, u}(x — m, v} d(i(x),

A group G G is representable in a Banach space X X if G G is isomorphic to the group of isometries on X X in some equivalent norm. We prove that a countable group G G is representable in a separable real Banach space X X in several general cases, including when G ≃ { − 1 , 1 } × H G \simeq \{-1,1\} \times H , H H finite and dim ⁡ X ≥ | H | \dim X \geq |H| , or when G G contains a normal subgroup with two elements and X X is of the form c 0 ( Y ) c_0(Y) or ℓ p ( Y ) \ell _p(Y) , 1 ≤ p > + ∞ 1 \leq p >+\infty . This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X X and a countable bounded group G G of isomorphisms on X X containing − I d -Id , there exists an equivalent norm on X X for which G G is equal to the group of isometries on X X . We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least 4 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

Introduction In this chapter we present the First Main Theorem. This theorem gives a criterion for determining which processes in analysis and physics preserve computability and which do not. A major portion of this chapter is devoted to applications. Here by the term “process” we mean a linear operator on a Banach space. The Banach space is endowed with a computability structure, as defined axiomatically in Chapter 2. Roughly speaking, the First Main Theorem asserts: bounded operators preserve computability, and unbounded operators do not. Although this already conveys the basic idea, it is useful to state the theorem with a bit more precision. The theorem involves a closed operator T from a Banach space X into a Banach space Y. We assume that Tacts effectively on an effective generating set {e n } for X. Then the conclusion is: T maps every computable element of its domain onto a computable element of Y if and only if T is bounded. We observe that there are three assumptions made in the above theorem: that T be closed, bounded or unbounded as the case may be, and that T acts effectively on an effective generating set. We now examine each of these assumptions in turn. Consider first the assumption that T be bounded/unbounded. This assumption is viewed classically; it has no recursion-theoretic content. In this respect, the approach given here represents a generalization and unification of that followed in Chapters 0 and 1. In Chapters 0 and 1, we gave explicit recursion-theoretic codings —a different one for each theorem. The First Main Theorem also involves a coding, bujt this coding is embedded once and for all in the proof. In the applications of the Fitst Main Theorem, no such coding is necessary. Thus, in these applications, we are free to regard the boundedness or unboundedness of the operator as a classical fact, and the effective content of this fact becomes irrelevant. Consider next the assumption that T be closed. The notion of a closed operator is standard in classical analysis. It is spelled out, with examples, in Section 1. However, if the reader is willing to assume the standard fact—that all of the basic operators of analysis and physics are closed—he or she could simply skip Section 1.

Introduction In this chapter we present the First Main Theorem. This theorem gives a criterion for determining which processes in analysis and physics preserve computability and which do not. A major portion of this chapter is devoted to applications. Here by the term “process” we mean a linear operator on a Banach space. The Banach space is endowed with a computability structure, as defined axiomatically in Chapter 2. Roughly speaking, the First Main Theorem asserts: bounded operators preserve computability, and unbounded operators do not. Although this already conveys the basic idea, it is useful to state the theorem with a bit more precision. The theorem involves a closed operator T from a Banach space X into a Banach space Y. We assume that Tacts effectively on an effective generating set {e n } for X. Then the conclusion is: T maps every computable element of its domain onto a computable element of Y if and only if T is bounded. We observe that there are three assumptions made in the above theorem: that T be closed, bounded or unbounded as the case may be, and that T acts effectively on an effective generating set. We now examine each of these assumptions in turn. Consider first the assumption that T be bounded/unbounded. This assumption is viewed classically; it has no recursion-theoretic content. In this respect, the approach given here represents a generalization and unification of that followed in Chapters 0 and 1. In Chapters 0 and 1, we gave explicit recursion-theoretic codings —a different one for each theorem. The First Main Theorem also involves a coding, bujt this coding is embedded once and for all in the proof. In the applications of the Fitst Main Theorem, no such coding is necessary. Thus, in these applications, we are free to regard the boundedness or unboundedness of the operator as a classical fact, and the effective content of this fact becomes irrelevant. Consider next the assumption that T be closed. The notion of a closed operator is standard in classical analysis. It is spelled out, with examples, in Section 1. However, if the reader is willing to assume the standard fact—that all of the basic operators of analysis and physics are closed—he or she could simply skip Section 1.

Let $X$ be a real reflexive and separable Banach space having the Kadeč-Klee property, compactly imbedded in the real Banach space $V$ and let $G\colon V\rightarrow {\mathbb R} $ be a differentiable functional. By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch [< i> Infinitely many solutions of a symmetric Dirichlet problem< /i> , Nonlinear Anal. < b> 20< /b> (1993), 1205–1216] and Bartsch-Willem [< i> On an elliptic equation with concave and convex nonlinearities< /i> , Proc. Amer. Math. Soc. < b> 123< /b> (1995), 3555–3561], respectively), we will study the multiplicity of solutions for operator equation $$ J_{\varphi}u=G^{\prime}(u), $$ where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Selfadjoint operators on real or complex Banach spaces

If H is any compact Hausdorff space-that is, a Hausdorff space in which any arbitrary covering of the space by open sets can be reduced to a covering by a finite number of them-the aggregate C(H) of all real-valued continuous functions x(r), where r runs over II, is of interest from any one of several viewpoints. C(H) is a real Banach space with the natural notions of addition of two of its members, and multiplication by a real scalar; the norm of any element X(T) is defined by the maximum of I X(7) I for r E H. It may also be considered as a commutative normed ring with scalars, the operation of multiplication being again the usual one of multiplication of two functions. And from a third point of view it is a linear lattice, the partial ordering in this case being the one which again naturally suggests itself: x(r) < y(r) if this relationship is true in its elementary sense for every r e H. From each of these three aspects the systems C(H) are quite special. Not every Banach space is equivalent, in the sense in which that term is naturally defined for such spaces, to such a system 0(H), and the same is true of rings and lattices. The question arises, then, to determine, in each of these three classes, which ones of their members are so representable.' In the case of rings and lattices the answer has been given: necessary and sufficient conditions are known in order that a system of either of these two kinds shall be equivalent, in a natural and specified sense, to such a system C(H). For the class of Banach spaces the question has received some attention2 but it is believed that no answer has as yet appeared.3 Our object in the present paper is to furnish an answer to this question: we determine necessary and sufficient conditions on a real Banach space B in order that it shall be a C-space-in other words, that there shall exist a compact Hausdorff space H such that B is equivalent to C(H) in the sense that it is mappable onto 0(H) by a linear isomorphism which preserves the norm. It should be noticed that the apparent gain in generality obtained by considering more general spaces is not a real one. We might consider instead of a compact H, for example, a more general topological space T-say a T, space;

A real Banach space E of dimension _3 is an inner product space iff there exists a bounded smooth convex subset of E which is the range of a nonexpansive retraction. De Figueiredo and Karlovitz [3] have shown that if E is a strictly convex real finite-dimensional Banach space and dim E> 3 then there can exist no bounded smooth nonexpansive retract of E unless E is a Hilbert space. (A subset F of E is a nonexpansive retract of E if it is the range of a nonexpansive retraction r: E-F.) This is a consequence of their more general result that if E is reflexive and a convex nonexpansive retract of E has at a boundary point xo a unique supporting hyperplane xo+H then H is the range of a projection of norm 1. As they have pointed out, the latter theorem fails in nonreflexive spaces (the unit ball of C[O, 1] furnishes a counterexample). Nevertheless, their first result is true in general: THEOREM. Suppose E is a real Banach space with dim E> 3. Then E is an inner product space iff there exists a bounded smooth nonexpansive retract of E with nonempty interior. We separate out of the proof of the theorem a lemma, valid in all real Banach spaces: LEMMA. Suppose F is a bounded smooth closed convex subset of a real Banach space E and F has nonempty interior. Then given disjoint bounded closed convex sets M and K in E with K compact, there exist p E E and 2>0 such that Kcp+)LF and (p+ 2F) rnM= 0. PROOF OF LEMMA. Clearly the hypotheses and conclusions of the lemma are invariant if K and M are translated by the same vector; thus without loss of generality we may assume 0 E K. Similarly, we may also assume 0 E int F. Since K is compact and M is closed, a basic separation theorem for convex sets assures the existence of a closed hyperplane H which strictly separates M and K; that is, there exist we E*, c e R' Received by the editors June 26, 1972 and, in revised form, August 21, 1973. AMS (MOS) subject classifications (1970). Primary 46C05.

Rao has firstly introduced the Riemannian structure associated with the Fisher information matrix over a finite dimensional parametrized statistical model. He proposed the Riemannian distance as a measure of dissimilality between two probability measures, (cf. [2], for example.) In [1], Amari introduced a pair of dual affine connections with respect to the metric and discussed of the differential geometry of the space of a finite dimensional parametrized statistical model. It provides a differential geometrical meaning to statistical inference. In the present paper, we realize the above idea for a family of equivalent (i.e.,mutually absolute continuous) Gaussian measures on a Banach space. Our main result is as follows. Let 5 b e a real separable Banach space and P be a centered gaussian measure on the topological dual of B (cf. [6]). The covariance of P naturally determines the Hubert space H. i.e., for arbitrary x1 and x2eB, let

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

Fernique's recent proof of finiteness of positive moments of the norm of a Banach-valued Gaussian random vector X is used to prove rth mean convergence of reproducing kernel series representationsof X. Embedding of the reproducing kernel Hilbert space into the Banach range of X is explicitly given by Bochner integration. This work extends and clarifies work of Kuelbs, Jain and Kallianpur. Fernique [2] has recently proved in a most elementary way that lima,<0+ E exp alIfX11%< oo for every centered Gaussian random vector X taking values in a real and separable Banach space B. As will be shown below, this result can be used to provide a dramatically simple proof of the strong convergence of certain representations of X by a series in B, as given by Kuelbs [4] and Jain-Kallianpur [3]. The role of reproducing kernel Hilbert spaces in such representations is sharply revealed by this approach. In this paper, B is a real and separable Banach space, B* its topological dual, X is the a-algebra generated by the open subsets of B, and P is a probability measure on a for which the induced distributions of the random variables x*CB* are all Gaussian with zero means. Suppose that in addition to being a Banach space, B is also a subset of the set of real functions on a set T (distinct points of B also being distinct as real functions on T), and that for each teT the evaluation mapping X, defined by XI(x)=x(t), xeB, is continuous on B. For example, if T is taken equal to B*, each xeB may be viewed as the continuous linear evaluation function on B* defined by x(x*)=x*(x), x*eB*. Let Y denote P quadratic-mean closure of {Xt, teT} viewed as a Hilbert subspace of Received by the editors February 6, 1971 and, in revised form, April 26, 1971. AMS 1970 classifications. Primary 60G15; Secondary 60G17.