3
votes
2answers
290 views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
1
vote
0answers
41 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
0
votes
0answers
31 views

Compact/open subsets of vector space

Let $K\subset U$ be compact (resp. open) subsets of a normed vector space. Must there exist an open $V$ such that $K+V\subset U$?
5
votes
1answer
80 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
0
votes
0answers
19 views

About Weakly Lindelöf Determined Banach spaces

I'd like to know where can I read more about weakly lindelöf determined (WLD) spaces. Especifically, I need to prove: 1.- Every weakly compactly generated is WLD 2.- If X is WLD then (X*,w*) is ...
0
votes
1answer
50 views

Compact subsets of $c_0$

Let $c_0$ be the Banach space of all sequences converging to 0, equipped with the supremum norm. How do the compact subsets of $c_0$ look like? I could imagine that $K \subset c_0$ is compact if ...
0
votes
1answer
88 views

Compact Operators and Complete Metrics Spaces

I have a couple of questions about compact operators and compactness in complete metric spaces: 1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is ...
1
vote
1answer
73 views

Weak* sequentiality

Suppose we are given a Banach space $E$ such that weak* compact subsets of $E^*$ are weak* sequentially compact (for example this happens when $E$ is separable). Does it follow that if $A$ is a subset ...
3
votes
1answer
744 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
0
votes
1answer
24 views

A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact

Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$ Does it follow that $C$ is pre-compact? In particular ...
1
vote
1answer
47 views

The set of finite “variations” of an unconditionally convergent series is pre-compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum_{i=1}^n \varepsilon_ix_i:n\in\mathbb N, \varepsilon_i=\pm1\}$ is pre-compact. Proof: 1) ...
1
vote
1answer
33 views

The set of “variations” of an unconditionally convergent series is compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum \varepsilon_ix_i:\varepsilon_i=\pm1\}$ is compact. Proof: 1) $\{-1,1\}^{\mathbb N}$ is ...
2
votes
1answer
256 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
368 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 ...
10
votes
1answer
186 views

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

Is there an analytic characterisation of the Čech-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
121 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 ...
1
vote
1answer
757 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
312 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
86 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
114 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 $$\|u_{n}\|_{\infty}\leq C$$ $C$ ...
5
votes
1answer
692 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
219 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
289 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 ...
8
votes
5answers
939 views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...