# Tagged Questions

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### On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
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### fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
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### cantor intersection theorem in banach space

Here is part of the question in my HW. Let$\ \{C_n\}\subset X$ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is ...
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### How to prove that the closed convex hull of a compact subset of a Banach space is compact?

Can anyone help me with this problem? Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form ...
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### Cancellation of addition on convex sets

I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space). I ...
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### Ball contained in a convex cone

Let $X$ be a Banach space. Let $C\subset X$ be a closed convex cone with nonempty interior. We denote by $B_X$ and $S_X$ the unit ball and the unit sphere of $X$ respectively. Let $e\in C\cap S_X$, ...
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### Convex function on Banach space

Let $(Y,\|\cdot\|)$ a Banach space and $b\colon Y\to \mathbb{R}$ a nonnegative convex function such that, for some $\mathcal{E}>0$, the set $\{y\in Y\,:\, b(y)<\mathcal{E}\}$ is nonempty and ...
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### Where to find information on the Hilbert cube in $\ell^2$

The Hilbert cube $H$ in $\ell^2=\ell^2(\mathbb{R})$ is the subset given by $$H=\lbrace(x_n)=(x_1,x_2,\ldots)\in\ell^2:|x_n|\le2^{-n} \text{ for }n=1,2,\ldots\rbrace.$$ I've heard that ...
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### Are countable intersections of convex sets convex?

Let $X$ be a Banach space $\{C_n\colon n\in\mathbb N\}$ a collection ofconvex sets in $X$. Is the set $$C=\bigcap_{n\in\mathbb N}C_n$$ convex?
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### Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1$} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
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### Hilbert space $H$ is strictly smooth

I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1$. To show this I think I should show $H$ is uniformly smooth first. ...
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### A question about the coercivity of a lsc and convex function.

I was doing a proof and I need to show a result to conclude it: $X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$. $f:X\to\overline{\mathbb{R}}$ is lower ...
I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...