Geometric duality theory of cones in dual pairs of vector spaces

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.

On Birkhoff–James orthogonality preservers between real non-isometric Banach spaces

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that if a dual Banach space $X$ is universally-left-stable, then it is isometric to a complemented $w^*$-closed subspace of $\ell_\infty(\Gamma)$ for some set $\Gamma$, hence, an injective space; and that a Banach space is universally-left-stable if and only if it is a cardinality injective space; and universally-left-stability spaces are invariant.

Selfadjoint operators on real or complex Banach spaces

Isometric reflection vectors in Banach spaces

There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

Introduction: One of the main difficulties-by no means the only onein the development of a theory of analytic functions in normed linear spaces stems from the fact that the modulus of a homogeneous polynomial and that of its polar are not always the same. We give here the theorenm that asserts that the modulus of a homogeneous polynomial and its polar are always equal in complex' Banach spaces. This has many important imi lications including the great simplifications brought about in the proofs of several furndamental theorems. The situation in real2 Banach spaces is quite different as we shall show. We base our proofs on a few fundamental lemmas in normed linear spaces: one generalizes Bernstein's theorem3 on the bounds and first derivative of S, a polynomial in the complex plane and two others generalize theorems of the brothers, A. Markoff4 and V. Markoff5 on the first and higher derivatives of polynomials of a real variable. These lemmas have an interest in themselves and have many applications not discussed in this paper. To present our results as briefly as possible, we assume familiarity with the Frechet differential calculus and analytic function theory in both real and complex Banach spaces. Theorems in Real Banach Spaces. The result of A. Markoff states that if pn(x) is a polynomial of degree n in a real variable x and if lPn(x)I < 1 in the interval xI < 1, then the derivative pn(x) satisfies the inequality lpA(x)l < nZ in |x| < 1. The following lelmma generalizes Markoff's result. Lemma 1. Let n

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = exp n +1τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .

In this paper, we consider a generalized system in real Banach spaces. Using Brouwer’s fixed-point theorem, we establish some existence theorems for generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity for a bifunction and extend Minty’s lemma for a generalized system. Furthermore, using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for a generalized system with monotonicity in real reflexive Banach spaces.

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.

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.

In this paper, we introduce a generalized system (for short, GS) in real Banach spaces. Using Brouwer’s fixed point theorem, we establish some existence theorems for the generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system. And using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces. As the continuation of existing studies, our paper present a series of extended results based on existing corresponding results.

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.

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces