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Results tagged with reference-request
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user 42411
This tag is used if a reference is needed in a paper or textbook on a specific result.
3
votes
1
answer
219
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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
$$\{e^*\in E^*: e^* \text{ attains its …
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0
votes
0
answers
65
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Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at …
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1
vote
1
answer
176
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Reference on vector-valued convex conjugate
The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot …
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2
votes
0
answers
93
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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
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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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1
vote
2
answers
212
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Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Sto...
Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space.
Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued b …
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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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7
votes
1
answer
434
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Reference Request: Existence of Ordinal Rank Theory?
Notations: Recall that $\omega_1$ is the first uncountable ordinal.
Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$
…
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2
votes
0
answers
65
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Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and …
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