All Questions
Tagged with sequences-and-series functional-analysis
1,110
questions
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55
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Is a linear map bounded if you can pull out a series?
Let $X,Y$ be Banach spaces and $A: X \rightarrow Y$ a linear mapping. I was wondering whether the following equivalence holds $$ A \quad \text{bounded} \Leftrightarrow A\left(\sum_{n=0}^\infty a_n\...
- 310
1
vote
1
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44
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Convergence of linear functionals
Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of linear functionals in $\mathrm{BV}^*$, namely the dual of the space of functions with bounded variation. Suppose that $\ell_n$ converges to $\ell$ as $...
- 1,136
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40
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Equivalence in $l_q$ spaces
Let $\theta \in (0,1)$ and $1\leq q<\infty$. Let $\lambda=(\lambda_n)$ a sequence of real (or complex) numbers such that $$||(2^{-n\theta}\lambda_n)||_q<\infty.$$ Now, let $\beta=(\beta_n)$ ...
2
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60
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If $f_n\to f$ a.e. and $\limsup_{n\to +\infty} \|f_n\|\le K,$ does it imply that $\|f\|$ is bounded?
Let $(H, \|\cdot\|)$ denote a Hilbert space which is continuously embedded in $L^2(\mathbb R^n)$. Let $\{f_n\}$ be a sequence such that
$$ f_n\to f \text{ a.e. in } \mathbb R^n,$$
and
$$\limsup_{n\to +...
- 425
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39
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Show that two subspaces $X$ and $Y$ of $\ell^1 (\mathbb{R})$ are such that $\overline{X+Y} = \ell^1 (\mathbb{R})$.
Let $E = \left(\ell^1 (\mathbb{R}),\lVert \cdot\rVert_1\right)$ and consider the subspaces
$$X = \left\{\left(x_n\right)_{n\in \mathbb{Z}_{>0}} \in E: x_{2n} = 0, \forall n\geq 1\right\},\hspace{...
- 280
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1
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38
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understanding the property of partition of unity [closed]
Let $Ω⊂R^n$ be open , $Ω=∪Ω_i$ , i∈I , $Ω_i$ open ⇒∃ {$φ_i$:i∈I} partition of unity such that
"support" ($φ_i$ )⊂$Ω_i$ ∀ i∈I .
{"support" ($φ_i$) ∶i∈I} is locally finite .
0≤$...
- 135
1
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26
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References on solid sequence spaces
I have been trying to search in different books on functional analysis for examples and properties of solid sequence spaces (https://en.wikipedia.org/wiki/Solid_set), however I cannot find any ...
- 659
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1
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34
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Finding the conjugate operator of the following operator
Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k} $ , where $y^0 \in l^2$ — fixed element.
Show, that conjugate operator $A^*$ exists and find it. Show, ...
- 21
4
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164
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dual space of l2 with strange norm
Consider $\displaystyle(\ell_2, \lVert\cdot\rVert_\star), \lVert x\rVert_\star = \sum\limits_{k=1}^{\infty}\frac{|x(k)|}{k}$. What is its dual space? Is this space reflexive?
My idea is to consider $\...
- 139
2
votes
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answers
46
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The sequence space cs has the Fatou property
The sequence space cs is defined as the space of all complex or real sequences $x=\left \{ x_n \right \}_{n\in \mathbb{N}}$, such that $\sum_{k=0}^{\infty} x_k$ converges.
I was wondering if this ...
- 115
2
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answers
45
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Is this infinite sum of matrices convergent?
For a matrix $A$, by $||A||$, I mean the matrix-norm induced by the $\ell^2$-norm. Let $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times p}$, $C\in\mathbb{R}^{n\times m}$ with $||A|| < 1$. ...
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23
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Sequence of sequences $\{a^{(n)}\}_n \subseteq \ell^2$ with bounded members $|a^{(n)}_k| \leq 1$ has converging subseq.
I struggling to understand a partial step in the solution to an exercise:
Given a seq. of seq. $\{a^{(n)}\}_{n \in \mathbb N} \subseteq \ell^2$ such
that $|a^{(n)}_k| \leq 1 \forall n,k \in \mathbb N$...
- 139
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1
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59
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Density argument in Normed spaces
Assume $(X,\|\cdot\|_X)$ is a normed space and if $Y$ is a dense linear subspace of $X$,
How to prove that for each $x \in X$ there exists a sequence $(y_j)_j \subset Y$ such that
$$\sum_{j=1}^\infty ...
- 24.2k
1
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1
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36
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Assuming that $(f_n)_n$ is bounded, does $\sup_{n\in\mathbb N}||f_n||_H <+\infty$ hold?
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.
I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}\|f_n\|_H <+\infty.$$
I'...
- 425
1
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1
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69
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Does a function having the property that for every $a_n\rightarrow a$ $f(a_n)$ diverges exist?
Could you help me solve the following tricky exam problem? Let $f: [
-1, 1]\rightarrow \mathbb R$. Let $a\in [-1, 1]$. Suppose that for every not ultimately constant sequence {$a_n$}, $a_n\in [-1, 1]$,...
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