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Results tagged with banach-spaces
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user 42411
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
3
votes
1
answer
160
views
Is it true that every Banach space has at least one extreme point that is normed by some point?
Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$
Denote
$$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$ …
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0
votes
1
answer
114
views
Does there exists an extreme point $(a_1^*,...,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0...
Fix a natural number $n\geq 1.$
Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying
$$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$
Let
$$B_{\mu} = \{(a_1,...,a_n)\in \m …
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3
votes
1
answer
219
views
Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?
Recall the Bishop-Phelps Theorem.
Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$
Then the set
$$\{e^*\in E^*: e^* \text{ attains its …
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8
votes
2
answers
601
views
If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?
Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E …
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0
votes
0
answers
65
views
Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at …
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0
votes
0
answers
115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\a...
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\ …
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1
vote
1
answer
176
views
Reference on vector-valued convex conjugate
The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot …
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2
votes
0
answers
93
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which i …
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1
vote
0
answers
213
views
Status of an open problem in isometric aspect of Banach space theory
The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$
Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach …
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7
votes
0
answers
323
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Status of two Banach space theory open problems posted by Pełczyński
In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems.
Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is …
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4
votes
2
answers
1k
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If the closed unit ball of Banach space has at least one extreme point, must the Banach spac...
Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am intereste …
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