Separation and Weak König's Lemma

1999 ◽  
Vol 64 (1) ◽  
pp. 268-278 ◽  
Author(s):  
A. James Humphreys ◽  
Stephen G. Simpson
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Separation Theorem ◽  
Space Theory ◽  
Open Convex ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractWe continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals A convex-valued selection theorem with a non-separable Banach space

2018 ◽  
Vol 7 (2) ◽  
pp. 197-209
Author(s):  
Pascal Gourdel ◽  
Nadia Mâagli
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Separable Banach Space ◽  
Lower Semicontinuous ◽  
Constructive Proof ◽  
Selection Theorem ◽  
Finite Dimensional ◽  
Nonempty Convex

AbstractIn the spirit of Michael’s selection theorem [6, Theorem 3.1”’], we consider a nonempty convex-valued lower semicontinuous correspondence {\varphi:X\to 2^{Y}}. We prove that if φ has either closed or finite-dimensional images, then there admits a continuous single-valued selection, where X is a metric space and Y is a Banach space. We provide a geometric and constructive proof of our main result based on the concept of peeling introduced in this paper.

Almost-Lp-projections and Lp isomorphisms

1989 ◽  
Vol 113 (1-2) ◽  
pp. 13-25 ◽  
Author(s):  
Michael Cambern ◽  
Krzysztof Jarosz ◽  
Georg Wodinski
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Case ◽  
Geometric Condition ◽  
Finite Dimensional ◽  
Measure Spaces ◽  
Finite Dimensional Case

SynopsisLp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.

scholarly journals Banach Spaces of Almost Universal Complemented Disposition

2019 ◽  
Vol 71 (1) ◽  
pp. 139-174
Author(s):  
Jesús M F Castillo ◽  
Yolanda Moreno
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Complemented Subspace ◽  
Finite Dimensional ◽  
Dimensional Decomposition ◽  
Finite Dimensional Decomposition ◽  
Separable Spaces

Abstract We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

TRANSLATING FINITE SETS INTO CONVEX SETS

2001 ◽  
Vol 33 (6) ◽  
pp. 711-714 ◽  
Author(s):  
EVA MATOUšKOVA
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Small Diameter ◽  
Empty Interior ◽  
Finite Sets ◽  
Bounded Set ◽  
Finite Set

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

scholarly journals Phelps spaces and finite dimensional decompositions

1988 ◽  
Vol 37 (2) ◽  
pp. 263-271 ◽  
Author(s):  
R. Deville ◽  
G. Godefroy ◽  
D.E.G. Hare ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Continuity Property ◽  
Finite Dimensional ◽  
Point Of Continuity Property ◽  
Convex Point

We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

scholarly journals An Example Concerning Exposed Points

1965 ◽  
Vol 8 (3) ◽  
pp. 323-327
Author(s):  
M. Edelstein
Keyword(s):  
Banach Space ◽  
Convex Body ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Direct Construction ◽  
Bounded Convex

In his paper [1] V. L. Klee gave an example of a smooth bounded convex body C, in En, with the property that extC s closed and extC - expC is dense in extC. As in [1] extC and expC denote the sets of extreme and exposed points of C respectively. It is the purpose of this note to exhibit a similar example in a general separable Banach space using a direct construction which involves Minkowski summation of convex sets.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals The Brouwer invariance theorems in reverse mathematics

2020 ◽  
Vol 8 ◽  
Author(s):  
Takayuki Kihara
Keyword(s):  
Reverse Mathematics ◽  
Base System ◽  
Open Problems ◽  
Explicit Algorithm ◽  
Topological Embedding ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

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