All Questions
Tagged with functional-analysis sequences-and-series
1,126 questions
1
vote
1
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50
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Show that the set $\{(x_k \cdot 2^{-k}) \mid x \in \ell_2\}$ is dense in $\ell_2$
I want to show that the set $A := \{(x_k \cdot 2^{-k}) \mid (x_k) \in \ell_2\}$ is dense in $\ell_2$.
My first attempt was that for any $(y_k) \in \ell_2$ also the sequence $(z_k) := (2^m \cdot y_k)$ ...
- 57
0
votes
0
answers
36
views
Sum of cosines in generalized functions
I've came across such a task: prove that this equality holds (in generalized functions)
$$\sum_{n = 1}^{\infty}{cos(nx)} = -\frac{1}{2} + \pi\sum_{k \in \mathbb{Z}}{\delta_{2\pi k}}$$
I was able to ...
- 9
0
votes
0
answers
56
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Convergence of $\cos(A)$ for operator $A$ in Banach space [closed]
For which linear bounded operators $A$ in Banach Space the following series converges in norm: $$\cos(A) = \lim_{n \to \infty}\sum_{k = 0}^{n}(-1)^k \frac{A^{2k}}{(2k)!}$$
I think it can be solved if ...
- 9
4
votes
1
answer
67
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Convexity of domain of unconditional convergence for power series in Banach algebras
Let $X$ be a Banach algebra. Consider the power series $\sum_{k=0}^\infty c_k x^k$. Let $D$ denote the set of elements in $X$ where this power series converges unconditionally (not merely the ...
- 1,095
3
votes
1
answer
92
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Little help in Rudin' s Functional Analysis 3.22
If $0<p<1$, show that $l^p$ contains a compact set $K$ whose convex hull is unbounded.
Attempt
Consider the sequence $x_n$ defined by $x_n(n) = n^{p-1}$ and $x_n(m) = 0$ if $m \neq n$ and the ...
- 133
0
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1
answer
62
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A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 39
5
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1
answer
126
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Hölder's inequality - conjugate exponent condition
For given $p\in[1,\infty]$, we define the sequence spaces
$$
\ell^p:=\{x=(x_n)_{n\in\mathbb{N}}\subset\mathbb{R}:\|x\|_p<\infty\}
$$
where
$$
\|x\|_p =\begin{cases}\left(\sum_{n=1}^\infty |x_n|^p\...
- 435
2
votes
2
answers
29
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Constructing a subsequence of $\{ f_k\} \subseteq L^p(\Omega)$ that satisfies these two properties
Let $1 <p< \infty$ and $\Omega \subseteq \mathbb{R}^n$. Assume that $T : D(T) \subseteq L^p(\Omega) \to L^p(\Omega)$ is a linear operator defined on a subset $D(T)$ of $L^p(\Omega)$. Assume that ...
- 819
1
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1
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55
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Compactness of a space of sequences
I consider the set $X=\{ (x_n)_{n\in\mathbb{N}} : \forall n\geq 0, x_n\geq 0,\;\sum_{n\geq 0}x_{n}\leq 1 \}$. I would like to show that $(X,d)$ where
$$
d(x,y) = \sum_{n\geq 0}\beta^n\lvert x_n - y_n\...
- 1,641
2
votes
1
answer
53
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Finding an open ball set
I have $X = (0, \infty)$ and $d(x,y) = |\frac{1}{x} - \frac{1}{y}|$
I want to find the open ball of radius $r>0$ centered at $a \in X$, so I must find $B_r(a) =$ {$x\in X: d(x,a) < r$} = {$x\in ...
- 49
2
votes
1
answer
68
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Limit of sequence in $C_b(\mathbb{R})$ [closed]
Consider $C_b(\mathbb{R}) = \{f| f:\mathbb{R}\rightarrow \mathbb{R}, f \text{ is continuous and bounded}\}$ with norm $\lVert\cdot\rVert_\infty$. For $f\in C_b(\mathbb{R})$ define $\displaystyle g_n(x)...
- 161
2
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0
answers
57
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Does $\frac{1}{n}\sum_{i=1}^n f_n(x_i) \to \frac{1}{|\Omega|}\int_\Omega f$ in some sense if $f_n \rightharpoonup f$ weakly in $L^2(\Omega)$?
I have a weakly converging sequence $f_n \rightharpoonup f$ in $L^2(\Omega)$ on a bounded domain $\Omega$ and I know $f_n, f \in C^0(\bar\Omega)$. Define the Monte Carlo estimator function $$R_n(g) := ...
- 704
1
vote
2
answers
60
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Limit of sequence of non-negative, $C^0 \cap H^1$ functions that are zero at more and more points
I have a sequence $\{f_n\}$ of functions such that for each $n$, we got
$f_n \in H^1(\Omega) \cap C^0(\bar\Omega)$
$f_n \geq 0$
$\lVert f_n \rVert_{H^1(\Omega)} \leq C$ for a constant $C>0$ ...
- 475
5
votes
1
answer
189
views
Taking the $n$th power of an infinite dimensional matrix
I have a certain matrix $$M:=\begin{bmatrix}
0&-2&0&0&0\\
1&0&-4&0&0\\
0&1&0&-6&0\\
0&0&1&0&-8\\
0&0&0&1&0\\
&&&...
- 327
1
vote
1
answer
89
views
Trouble finding the general solution to Volterra's integral equation
Solve the Volterra integral equation for $\lambda = 1$
$$ x(t)=\int_0^t (t-s) \cdot x(s) d s+t^3 $$
I am solving it by iterative approximations
$$
K_{n+1}(t, s)=\int_s^t K\left(t, {\tau}\right) K_{n+...
- 425
1
vote
0
answers
93
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There exists a set of complex numbers $\alpha_n:n\in\mathbb{Z^*}$ such that $x=\sum_{n\in\mathbb{Z^*}}\alpha_ne^{nx}$?
Note this is an infinite series! I have the same question for $1$, is it possible to have $1=\sum_{n\in\mathbb{Z^*}}\beta_ne^{nx}$ for some complex coeficients $\beta_n\in\mathbb{C}$?
I am considering ...
- 141
2
votes
0
answers
102
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Convergence rate of Laguerre coefficients for polynomially bounded functions
Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies:
$$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$
for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
- 667
0
votes
1
answer
64
views
Is a linear map bounded if you can pull out a series?
Let $X,Y$ be Banach spaces and $A: X \rightarrow Y$ a linear mapping. I was wondering whether the following equivalence holds $$ A \quad \text{bounded} \Leftrightarrow A\left(\sum_{n=0}^\infty a_n\...
- 423
1
vote
1
answer
57
views
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 $...
- 1,156
1
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0
answers
42
views
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
votes
0
answers
62
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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
$$\limsup_{n\to +...
- 425
0
votes
1
answer
40
views
Show that two subspaces $X$ and $Y$ of $\ell^1 (\mathbb{R})$ are such that $\overline{X+Y} = \ell^1 (\mathbb{R})$.
Let $E = \left(\ell^1 (\mathbb{R}),\lVert \cdot\rVert_1\right)$ and consider the subspaces
$$X = \left\{\left(x_n\right)_{n\in \mathbb{Z}_{>0}} \in E: x_{2n} = 0, \forall n\geq 1\right\},\hspace{...
- 314
0
votes
1
answer
43
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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≤$...
- 191
1
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0
answers
28
views
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 ...
- 669
1
vote
1
answer
37
views
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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1
answer
227
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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 $\...
- 161
2
votes
0
answers
47
views
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 ...
- 105
2
votes
0
answers
48
views
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 $||A|| < 1$. ...
0
votes
1
answer
25
views
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
0
votes
1
answer
63
views
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
$$\sum_{j=1}^\infty ...
- 24.5k
1
vote
1
answer
40
views
Assuming that $(f_n)_n$ is bounded, does $\sup_{n\in\mathbb N}||f_n||_H <+\infty$ hold?
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.
I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}\|f_n\|_H <+\infty.$$
I'...
- 425
2
votes
1
answer
81
views
Does a function having the property that for every $a_n\rightarrow a$ $f(a_n)$ diverges exist?
Could you help me solve the following tricky exam problem? Let $f: [
-1, 1]\rightarrow \mathbb R$. Let $a\in [-1, 1]$. Suppose that for every not ultimately constant sequence {$a_n$}, $a_n\in [-1, 1]$,...
- 303
1
vote
1
answer
58
views
Does the identity $ I(u) = c$ hold?
Let $(H, ||\cdot||)$ be a Hilbert space and let $I\in C^1 (H, \mathbb R)$.
Assume that $(u_n)_n\subset H$ is a bounded sequence in $H$ and let $u$ its weak limit.
Also, suppose that
$$\lim_{n\to +\...
user603537
1
vote
1
answer
145
views
If a sequence is bounded, does its weak limit happen to be bounded?
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$, i.e. there exists $C>0$ such that
$$\|f_n\|\le C \quad\forall n.$$
Let $f$ denote the weak limit of $f_n$.
...
- 425
0
votes
1
answer
62
views
Existence and uniqueness of solution for the same problem
In a programming course, students are tasked with writing a code to find $n$ points that satisfy the following conditions:
$0=x_0<x_1<...<x_n=1$
$\displaystyle {\sum_{1\leq j\leq n}}_{j\...
- 11
1
vote
1
answer
73
views
Determining if infinite series of increasing derivatives is a continuous functional
Let $D(\mathbb{R})$ be the space of test functions on $\mathbb{R}$, i.e. the set of smooth compactly supported functions on $\mathbb{R}$, and where we say $\phi_k \to \phi$ if $\phi_k$ and all its ...
- 388
1
vote
1
answer
93
views
Trouble verifying a step in a proof regarding the span of an orthonormal family
I'm reading Sheldon Axler's Measure, Integration & Real Analysis and I am confused about a step that the reader is asked to verify. The theorem is:
Suppose $\left\{e_k\right\}_{k \in \Gamma}$ is ...
- 99
2
votes
2
answers
243
views
Exercise on sequence of a function
Subject: Seeking Help with a Mathematics Exercise
Hello everyone,
I hope this message finds you well. I'm reaching out to seek assistance with a mathematics exercise that has been posing some ...
- 145
1
vote
1
answer
101
views
Brezis' exercise 8.28.11: how to prove $\sum_{k=0}^{\infty} \left | \frac{1}{2} \alpha_k (f) - a \right |^2 < + \infty$?
Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by
$$
(Tf) (x) = \int_0^x t ...
- 17.9k
1
vote
1
answer
65
views
Maximum of $\ell^2$-Norm
For $r,c>0$ put
$$X_{c,r}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r, \, \forall i\in \mathbb{N}: |x_i|<c\}.$$
Then I can show that $\inf_{x \in X_{c, r}} \|x\|_2=0$.
Is it possible to compute
$$...
- 101
3
votes
0
answers
76
views
Making sense of generating functions in terms of the shift operator
While studying transforms, I stumbled across a neat way to express sequences, and have a couple of questions about its validity and how it connects to other topics. Suppose we are working with ...
- 358
2
votes
0
answers
86
views
Exercise 3.19 in Brezis' Functional Analysis
Exercise 3.19:
Let $E = \ell^p$ and $F = \ell^q$ with $1 < p < \infty$ and $1 < q < \infty.$ Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function such that
$$|a(t)| \leq C|t|^{\frac{p}...
- 79
1
vote
1
answer
68
views
Bounded sequence of real numbers and convergent subsequence
Let $(|x_{n}(\alpha_{k}^n)|)_n$ be a bounded sequence of real numbers , for each $k \in \mathbb{N}$ (meaning $|x_{n}(\alpha_{k}^n)| \leq M$ for one constant $M$ and all $n,k$) such that for fixed $n,$...
- 340
2
votes
0
answers
114
views
A question about probability theory and a little functional analysis
Consider the sequences of mean zero random variables $(X_n)_{n \in \mathbb N}$ with finite variances, i.e., $E[X_n]=0$ and $E\left[X_n^2\right] =: \sigma_n< \infty$, for each $n$. Moreover, ...
- 127
1
vote
1
answer
83
views
A question about the $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$ norms in $\ell^2$
I'm a little curious to understand the $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$ norms a little better.
The following example presents a sequence $(x_n)_{n \in \mathbb N}$ in $\ell^2$ that is in the ...
- 127
0
votes
1
answer
35
views
The space $c_{00}$ is direct sum of $M$ and $N$ but projections are not continuous
I don't know how to finish this problem because I don't know how to write the projections. If there is an alternative approach I would like to know it too (I suspect there may be one because $||\cdot||...
- 1,364
3
votes
1
answer
137
views
Is every sequence $\{x_n\}_{n=1}^\infty$ such that $\lim_{n\to\infty}x_n=0$ a member of $\ell^p$ for some $p>0$?
For every sequence of real numbers $(x_n)_{n\in N}$ converging to zero, does there exist a positive number p such that the sum $\sum_{n=1}^\infty |x_n|^p$ converges.
I'm not 100% sure if this is ...
- 305
4
votes
2
answers
101
views
Let $T\in M_m(\mathbb{C})$ with $r(T)=1$. Is it true that $\lim \frac{\lVert T^{n+1}\rVert}{\lVert T^n\rVert}=1$?
Here the norm $\lVert\cdot\rVert$ denotes the operator norm. If $T$ is a Jordan matrix with spectral radius $r(T)=1$, then I am able to prove that $\frac{\lVert T^{n+1}\rVert}{\lVert T^n\rVert}\to1$. ...
- 1,156
0
votes
1
answer
72
views
Proof by contradiction that $(\ell^p)^* \subseteq \ell^q$
For $p \in (1, \infty)$, let $\ell^p = \{(x_n) : \sum |x_n|^p < \infty\}$ and let $(\ell^p)^*$ denote the dual space of bounded linear forms on $\ell^p$ $(\mathbb{F} = \mathbb{R})$.
It is ...
- 2,495
5
votes
0
answers
236
views
Unconditional convergence implies convergence of sum of norm squares in hilbert space
Let $(H, (.,.))$ be a Hilbertspace and consider the definition of unconditional convergence, i.e. for a set $I \neq \emptyset$ and a family of vectors $(x_i)_{i \in I}$ in $H$ the series
$$ \sum_{i \...
- 368