0
votes
1answer
19 views

Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. [duplicate]

Define $A(K) = \{f : K \rightarrow \mathbb{R}$ $| f$ is continuous$\}$. $K$ is compact. Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. Since a Banach space is complete then every Cauchy ...
3
votes
2answers
253 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, ...
0
votes
0answers
34 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
1
vote
0answers
38 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 ...
4
votes
0answers
48 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
0
votes
0answers
18 views

About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
6
votes
1answer
54 views

Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the Arzelà–Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
2
votes
1answer
30 views

Arzela-Ascoli and compactness in $C(X), l^p, L^p$

Arzela-Ascoli and compactness in $C(X), l^p, L^p$ $C(X)$ with the uniform norm and $X$ is a compact metric space, a closed and bounded set in $C(X)$ is compact if and only if it is ...
2
votes
1answer
52 views

The set $\{\|f\|_\alpha \leq 1 \}$ has compact closure in $C([0,1])$

Recall the Holder norm $(0<\alpha\leq 1) $ $$\|f\|_\alpha = \max\bigg\{ |f(x)| + \frac{|f(x) - f(y)|}{|x-y|^\alpha} : x,y \in [0,1], x\neq y\bigg\}$$ I want to show that the set ...
0
votes
1answer
46 views

Boundedness of continuous functions on compact sets

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum. What happens if we replace $K$ by a closed ...
1
vote
1answer
35 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
4
votes
1answer
51 views

Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
0
votes
0answers
16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
3
votes
1answer
17 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
1
vote
1answer
38 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
2
votes
1answer
48 views

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Let $f_n \to f$ on compact subsets of the real line. If $u_m \rightharpoonup u$ in $L^2(0,T;H^1) \cap L^p(0,T;L^p)$ and $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in ...
5
votes
2answers
46 views

Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
2
votes
1answer
63 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
1
vote
1answer
79 views

why compact support implies a function vanished at boundaries?

"A function has compact support if its support is a compact set." While support of a function $u:G\rightarrow\mathbb{R}$ is defined to be $supp(u)=\overline{\{x:G|u(x)\neq0\}}$ But lately, Another ...
1
vote
3answers
50 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
1
vote
0answers
42 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
3
votes
1answer
48 views

Proof of compactness of Lipschitz functions

Consider the set $\mathcal{F}$ of continuous functions on $[0;1]$ with boundary values $$ f(0)=f(1)=0 \qquad \forall f \in \mathcal{F}. $$ Define the metric $d(f,g) = \lVert f-g \rVert_\infty = ...
3
votes
2answers
210 views

Riesz's Theorem of compactness

$\left(X,\|\cdot\|\right)$ is a normed vector space. $\textbf{Riesz's Theorem of compactness}$ says that $$ \{x \in X \colon \|x\| \leq 1 \} \ \text{compact} \ \Longleftrightarrow \ \text{Each bounded ...
1
vote
0answers
55 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
-1
votes
1answer
31 views

Compactness Invariant between normed spaces

Let $X$ and $Y$ be finite dimensional normed spaces. Let $D:\X \rightarrow Y$ be an isometric isomorphism then if $X$ is compact the $Y$ is also compact. I have started by choosing a sequence in $Y$ ...
5
votes
1answer
78 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 ...
1
vote
1answer
25 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
1
vote
1answer
32 views

A compactness argument for small high frequencies

I would like to prove the following statement: Let $N\geq 1$, $1\leq q<\infty$ and let be $E$ a relatively compact subset of $L^q(\mathbb{R}^N)$. Then \begin{equation*} \sup_{u\in ...
1
vote
1answer
58 views

Strong convergence of bounded sequences in Bochner spaces

Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$. Suppose we have a sequence ...
1
vote
1answer
56 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
1
vote
0answers
29 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
1
vote
1answer
75 views

Locally-compact function spaces?

I ask this question out of curiosity, not a specific need. Euclidean spaces and manifolds. Are there examples of locally compact function spaces? Could (some?) Sobolev spaces be locally compact?
3
votes
2answers
52 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
2
votes
4answers
165 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
0
votes
1answer
63 views

Noncompactness of the closed unit ball in $L^2$

Let $$ L^2[0,1]=\{f:[0,1]\to\mathbb R\,\,\text{such that}\,\, \|f\|_2<∞\}, $$ where $\|f\|_2^2=\int_0^1 |f(x)|^2\,dx.$ Show that the unit sphere $$ S=\{f\in L^2[0,1]:\|f\|_2\le 1\}, $$ is ...
1
vote
1answer
33 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
1
vote
1answer
332 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
2
votes
1answer
69 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
2
votes
1answer
54 views

Compactness in $\mathbb{R}^{X}$

I'm reading a book chapter on weak topology, where the author identified the collection of all real functions on an abstract space $X$ with $\mathbb{R}^{X}$. I find it difficult to make sense out of ...
0
votes
2answers
81 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
3
votes
2answers
102 views

Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
1
vote
2answers
43 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
2
votes
1answer
109 views

Arzela-Ascoli net question

Let $X$ be a compact metric space. Let $C(X)$ denote the space of real-valued continuous functions on $X$. A commonly given corollary to the Arzela-Ascoli theorem is: Proposition: If $f_n$ is an ...
0
votes
1answer
233 views

$X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
3
votes
2answers
36 views

showing compactness for a subset of a function space

Our professor told us the following in lecture: Let $X_A:=\{f\colon A\to\mathbb R|f(A) \textrm{ is bounded}\}$ and $\alpha(f,g):=\sup\{|f(x)-g(x)| \;|x\in A\}$. Given $\beta\colon A\to\mathbb R, ...
3
votes
0answers
162 views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
0
votes
1answer
82 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 ...
3
votes
0answers
103 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
2
votes
1answer
115 views

The Arzelà–Ascoli theorem fails on a half-open interval

Can we find an example: (1) $\lbrace f_n \rbrace_n$ is a family of real-valued functions defined on $[0,1)$ such that this family is uniformly bounded and equicontinuous, $f_n(0)=0$; ~~~ Uniformly ...
1
vote
1answer
56 views

Is this a totally bounded set in the space of continuous functions?

If $A=\{f\in C[0, 1]: \int^1_0|f(x)|^2\,dx\leq1\}$ and metric $d(f, g)$ is $(\int^1_0|f(x)-g(x)|^2dx)^\frac{1}{2}$. Is $A$ totally bounded? I know $A$ is clearly bounded since $d(f, 0)\leq 1$ under ...