Tagged Questions

0
votes
0answers
39 views

Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
1
vote
1answer
89 views

Compact subspace of a Banach space .

The following statement doesn't make sense to me, can someone justify it to me ? If $K$ is a compact subset of a Banach space $Y$ then there exists for $\epsilon > 0 $ a finite dimensional ...
9
votes
1answer
155 views

Growth $\beta X\setminus X$ of a Banach space $X$

Is there an analytic characterisation of the Cech-Stone compactification (in the norm topology, which is a normal space) of a Banach space $X$? The reason I ask is because I want to know what the ...
1
vote
1answer
83 views

Weak compactness

Define a map $\varphi \colon [0,1]\to C[0,1]^*$ by $\varphi(x) = \delta_x$. Then $\varphi$ is a homeomorphism for the w*-topology. Let $K$ denote the image of $\varphi$. I have two questions: 1) Is ...
0
votes
1answer
294 views

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 ...
2
votes
1answer
119 views

How to prove that the sum of two compact sets in a Banach space need not be compact

Let $X$ be a Banach space and $K$ a compact subset of $X$ and consider for a given $\eta>0$ the closed ball $C(0,\eta)$ centered at $0$ of radius $\eta$. How can I show that $K+C(0,\eta)=\{x+y: ...
3
votes
2answers
63 views

How to show that this union is relatively compact using total boundedness

Edit I've made a mistake in the formulation. There should be in inclusion, not an equality. Let $E$ be a Banach space. Let $\varnothing\neq K_j\subset E$ be compact for all $j\ge1$ such that ...
5
votes
1answer
81 views

Compact Immersion $L^{p}\hookrightarrow L^{q}$

I was studying Evans book (Partial Differential Equations) and in page 279 he use the fact that if a sequence $u_{n}\in L^{\infty}(\mathbb{R}^{n})$ is such that $$\|u_{n}\|_{\infty}\leq C$$ $C$ ...
4
votes
1answer
227 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? 1) $X$ is reflexive; 2) $B$, the unit ball of $X$, is weakly compact.
4
votes
1answer
150 views

Compactness of unit ball in $\ell_\infty$ with a different norm

Consider the normed spaces (over the field of real numbers) $X=(\ell_\infty,\|\cdot\|_\infty)$ and $Y=(\ell_\infty,\|\cdot\|)$ where $$\|x\|=\sup_{n\in\mathbf{N}}\frac{|x_n|}{2^n}.$$ How can I show ...
2
votes
1answer
181 views

Are compacta in a complete infinite dimensional normed space nowhere dense?

Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense. I've been able to prove the statement for ...