All Questions
Tagged with lp-spaces compactness
86
questions
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32
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Convergence on $L^1$ space
I come across this problem while reading hints for Exercise 3.15 (Brezis):
Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that:
$$\displaystyle \int_{\Omega} \varphi f_n\...
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0
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44
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Trying to understand the proof for the criterion of compactness in $l_p$ space
I have the following theorem about the criterion of compactness in $l_p$ space
For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions
are equivalent:
i) $K$-totally bounded in $(l_p,||.||...
- 249
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How does $f$ being uniformly continuous imply $f_\epsilon \rightarrow f$ uniformly?
Let $j(x)$ be any positive infinitely differentiable function with support in $(-1, 1)$ so that
$$\int_{-\infty}^\infty j(x) dx = 1.$$
Define
$$j_\epsilon(x) = \epsilon^{-1} j(x/\epsilon).$$ If $f \in ...
- 6,275
2
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1
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94
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$L^1$ equicontinuity for functions with uniformly bounded total variation
I've been trying to prove that the space $BV[0, 1]$ of elements of $L^1 [0, 1]$ with bounded variation, is compactly embedded in $L^1[0, 1]$, using the Fréchet-Kolmogorov theorem.
For context, a ...
- 186
2
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1
answer
43
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For which values of the parameter $r$ the operator is compact?
I have an operator
$$A : L_{2}[0,1] \to L_{2}[0,1]$$ which is $$(Ax)(t)={t^{r-1}}\int_{0}^{t}\frac{x(s)}{s^r}ds$$
I already proved that operator A is bounded in $L_{p}[0,1]$ space when $p\in(1,\infty)$...
- 121
1
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120
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Possible Frechet-Kolmogorov compactness theorem equivalent in $L^\infty$?
The Frechet-Kolmogorov(-Sudakov) compactness theorem states that in $L^p(\mathbb{R}^n)$ for $1\leq p<+\infty$, a set of functions $\mathcal{F}$ is totally bounded if and only if $\mathcal{F}$ is
...
- 131
1
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1
answer
191
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Compact set on $L^p([0,1])$
I'm aware of Classifying the compact subsets of $L^p$ and have seen similar posts, but I haven't seen an example of a compact set on an $L^p$ space. Trying to think of a possible simple example, and ...
- 35
1
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29
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Intersection of compact sets in different spaces
Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both.
My idea:
Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger ...
- 71
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92
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Compact spaces in $l_p$ with $(1\leq p<+\infty)$
Let be $K \subset l_p$ closed set, with $(1\leq p<+\infty)$. Prove that $K$ is compact if and only if:
$\forall n\geq1:\sup\{|x_n| : x=\{x_n\}_{n\geq1}\}<+\infty$
$\lim_{m\rightarrow +\infty} \...
2
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2
answers
130
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Show that the set $M = \left\{ {x \in {\ell ^1},\left| {{x_k}} \right| \le \left| {{y_k}} \right|} \right\}$ is a compact subset of $\ell^1$?
Lets define the set $M = \left\{ {x \in {\ell ^1},\left| {{x_k}} \right| \le \left| {{y_k}} \right|} \right\}$ where $\ell^1$ is the set of sequences of infinite length with bounded $\ell^1$ norm, and ...
user830472
2
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116
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Linear span of a compact set
Is the linear span of a compact set in $L^2(X)$ complete?
More specifically, let $X$ be a locally compact Hausdorff space, and $E$ be a compact subset in $(L^2(X), ||.||_2)$. Then can we conclude that ...
- 330
3
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1
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244
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Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?
Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?
For $I: L^p([0,1]) \to C([0,1])$ with $p\in (1,\infty]$ this can be shown quite ...
- 365
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2
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151
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Subset of $\ell^2$ which is closed and bounded, but not compact [closed]
Consider the space $\ell^2=\left\lbrace x=(x_n)_{n\in \mathbb{N}}; \sum_{n=1}^{\infty}x_n^2 < \infty\right\rbrace$ with the inner product
$$\langle x,y \rangle = \sum_{n=1}^{\infty}x_n\cdot y_n$$
...
- 37
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0
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31
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Compact operators on $\ell^p$ [duplicate]
For any real sequence $(a_n)_{n\in \mathbb N}$ define the linear operator $A:\mathbb R ^\mathbb N \to \mathbb R ^\mathbb N$ by $Ax=(a_nx_n)_{n\in \mathbb N}$. Let $1\leq p\leq \infty$.
I know $A$ is a ...
- 281
1
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1
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43
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Can anyone give an example of a relatively compact set in $\ell_p$ which is not majorable? [closed]
Is there any relatively compact set in $\ell_p$ which is not majorable? If yes, can anyone give me an example?
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