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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\}.$ …

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 …

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 …

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 …

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 …

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 $\|\ …

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 …

Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$ It is easy to show that the following two statements are equivalent: $(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is …

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 …

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 …

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 …

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 …

Ian Morris quoted the following: For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\subs …

Based on the comment by Nik Weaver, the answer to my question is negative, that is, there exists a function $f:X \rightarrow \mathbb{R}$ such that it is not an infimum of any upper-semicontinuous func …

Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$. …