Questions tagged [functional-analysis]
Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).
53,998 questions
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Show that the inclusion $\phi$ from $l^1$ to $c_0$ is not a topological homomorphism
Let $l^1$ be the space of absolute summable sequences, that is $(x_n)_{n \in \mathbb{N}}$ such that $\forall n \in \mathbb{N}, x_n \in \mathbb{C}$ and $\sum_{n=1}^{\infty} |x_n| \lt \infty$.
Let $c_0$ ...
- 347
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59
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Proving $||Cesaro sum|| \leq ||f||$
How do I show the sup norm of the Cesaro sum of the Fourier partial sums is less than or equal to the sup norm of f i.e.
(Sorry I'm new to StackExchange) I have the integral form of the Cesaro sum ...
- 11
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1
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38
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Showing if operator $T-\lambda$ is bounded below for $|\lambda| <1$ then$\lambda \mapsto \dim (\mathcal{H}\ominus (T-\lambda)\mathcal{H})$ is constant
Here's what I am trying to show:
Let $\mathcal{H}$ be a separable Hilbert space. Let $T$ be bounded operator on $\mathcal{H}$ such that $T-\lambda$ is bounded below for each $\lambda \in \mathbb C$ ...
- 169
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51
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Proving an equivalent form of the essential spectrum for a self adjoint densely operator.
Let $T$ be a self adjoint densely defined operator on a Hilbert space. I am trying to show that a real number $\lambda$ will be in the essential spectrum iff there exists an orthonormal sequence $e_n$ ...
- 728
2
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2
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51
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Prove that there are no more numbers that belong to the spectrum or find them
Find the spectrum of the operator $ A $ in $L^2[0, \pi]$ if
$$(Ax)(t) = \int\limits_0^\pi \cos(t+2s)x(s) \, ds$$
Consider the operator $A$ in the space $L^2[0,\pi]$ given by the rule $(Ax)(t)=\int\...
- 1,541
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1
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48
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Bounding a quadratic form using projections
Let $\mathcal{H}$ be a Hilbert space and $T$ a positive linear operator, meaning that $\langle x, T x\rangle \ge 0$ holds for every $x \in \mathcal{H}$. Let $M\subset \mathcal{H}$ be a closed subspace,...
- 987
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Proving that a set is total in $l^{2}$
I have the following problem:
I have already done part a) of this question. I have tried to prove part b) by doing an $\epsilon$ argument but I couldn't. The definition of Schauder basis that I am ...
- 151
3
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answer
76
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Calculation of an operator norm $(Tf)(x)=x^2f(x)$
The question is given in the following way:
Let the operator $T:C[0,3]\rightarrow C[0,3]$ be defined by $$(Tf)(x)=x^2f(x), x\in [0,3]$$ If we equip $C[0,3]$ with the norm $\Vert f \Vert_{L^p[0,3]}=(\...
- 164
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Dual of $\ell_p$ for $0<p<1$
Let $0<p<1$. Prove that there is a one-to-one correspondence $\Lambda\leftrightarrow y$ between $(\ell^p)^*$ and $\ell^\infty$ given by
$$\Lambda (x) = \sum_{k=1}^\infty x(k)y(k)$$
Attempt
I am ...
- 133
1
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1
answer
32
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T/F: For $A_\lambda$ with $\lambda>0$ converging to $\cap A_\lambda$, $\sup_{f\in A_\lambda}|f(x)|$ converging to $\sup_{f\in \cap A_\lambda}|f(x)|$
Let $X$ be a Banach space and $f$ be a bounded linear functional defined on $X$. Consider a monotonically decreasing sequence $\{A_\lambda\}_\lambda$ with $\lambda>0$ such that $A_\lambda$ ...
- 560
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63
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Existence of norm 1 vector in Hilbert space that gives distance via inner product
I am trying to prove the following:
Let $H$ be a Hilbert space, $Y\subset H$ a closed subspace, and $x \in
H \backslash Y$. Then there exists a unique $z\in Y^\perp$ with $\| z
\| =1$ and $\langle ...
- 388
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an estimate $\|\partial_v w\|_{H^{-1/2}(\partial\Omega')}\leq C\|w\|_{H^1(\Omega)}$
$\partial_v w$ is conormal derivative of $w$.
Suppose that $\Omega$ is a Lipschitz domain and let $\Omega'$ be a compact subset of a $\Omega$.
Let $w:\Omega\rightarrow \mathbb{R}$ be a solution for a ...
- 303
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61
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Prove that $tr(T)=\sum _{\lambda \in \sigma_p(T)}\lambda \dim E_\lambda$
$T$ is a compact operator and a Trace-class operator on the Hilbert space $H$, prove that $tr(T)=\sum _{\lambda \in \sigma_p(T)}\lambda \dim E_\lambda$, where $E_\lambda =\ker(\lambda I-T)$.
The ...
- 534
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How to prove that Trace-class operators are Hilbert-Schmidt operators
Prove that if $T$ is a Trace-class operator on the Hilbert space $H$, then $T$ is also a Hilbert-Schmidt operator.
The definitions of Trace-class operators and Hilbert-Schmidt operators are as in ...
- 534
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1
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85
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Determining when the operator $f \mapsto if'$ is self adjoint with a specific domain.
Considering the Hilbert space for $L^2([0, 1])$ and $z \in \mathbb{C}$, we can define $T_z$ as the closure of $f \mapsto if'$ on the domain $\{f \in C^1([0, 1]): f(1) = zf(0)\}$. I think this domain ...
- 728
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67
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Unbounded operator on $L^2([a,b])$ with bounded inverse
Does there exist any unbounded operator $A$ on $Dom(A)\to L^2([a,b])$, with $Dom(A)\subseteq L^2([a,b])$ such that its range is dense (or onto) in $L^2([a,b])$ and it has an inverse defined on the ...
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"Is it rigorous to define an open ball in with the supremum norm? [closed]
I have the following definitions:
The open ball in the space of continuous functions on $[0,1]$, centered at the zero function and with radius $\|f\|$ (where $\|f\| > 0$ and the norm is the ...
- 27
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14
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Does the product of two reproducing kernels lie within the same RKHS?
Consider $k$ to be the reproducing kernel for the functions defined on the space $X\times Y$. Denote the $\mathcal{H}_{(X,Y)}$ as its RKHS. I would like to approximate a conditional expectation using ...
- 1,244
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30
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The Spectral Mapping Theorem for $\mathcal{C}^N$
I am trying to prove that $\sigma(p(T)) = p(\sigma(T))$ when $X = \mathcal{C}^N.$ I can see that $\sigma(p(T)) \subset p(\sigma(T))$ because if $z$ is an eigenvalue of $A$ then $p(z)$ is an eigenvalue ...
3
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1
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62
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Show that a $\sigma$-additive semi-norm is continuous.
Suppose that $E$ is a Banach space and $p$ is a $\sigma$-subadditive semi-norm, that is, if $\sum_{n=1}^{\infty}x_n$ converges in $E$, then
\begin{equation}
p(\sum_{n=1}^{\infty}x_n) \leq \sum_{i=1}^{\...
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-3
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0
answers
70
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The closure of $c_{00}$ in $\ell^1$ [closed]
Which is the closure of
$c_{00} = \{(x_n)_n \in \mathbb{R}: \exists N \in \mathbb{N}: \forall n \geq N: x_n = 0\}$ in $\ell^1(\mathbb{R})$?
I know that in $\ell^\infty(\mathbb{F})$ the closure of $C_{...
- 1
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1
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49
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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
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37
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Can we build a measure satisfying this constraint?
Let $\Sigma$ be a $\sigma$-algebra on some set $\Omega$. Note that $\Sigma$ might not be atomic and could even not contain no atoms at all.
Let $A\in\Sigma$, I want to prove that there exists a ...
- 6,032
1
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1
answer
24
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Linear functional on a Hilbert space defined by a sesquilinear form can be expressed in certain manner when restricting to a finite dim. subspace
Let $H$ be a Hilbert space and $a : H \times H \to \mathbb{K}$ where $\mathbb{K}$ is either complex or real numbers, a sesquilinear form such that there exist constants $C, c > 0$ such that
$$ |a(...
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32
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Density of a subset of Schwartz space in the p-fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $W^{s,p}(\mathbb{R}^N)$ ($0<s<1$ and $1<p<\infty$) defined as
$$
W^{s,p}(\mathbb{R}^...
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1
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70
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Loomis-Whitney inequality from generalized Hölder inequality
I've proven the following "generalized Hölder inequality",
Suppose $q^{-1}=\sum_{i=1}^np_i^{-1}$ for $1\leq q,p_1,...,p_n\leq\infty$, then $$||\prod_{i=1}^nf_i||_q\leq\prod_{i=1}^n||f_i||_{...
- 444
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1
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30
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Linear functionals on Von Neumann algebra tensor products
I am reading about the basics of Von Neumann algebra tensor products, and I am a bit confused on how one proves that the tensor product of bounded linear functionals on a VNA extends to the VNA tensor ...
- 1,777
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48
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The $\ell_p$ norm for $p=0.$ Why isn't the $p\to 0$ limit of the $p$-norm the product?
In the article on $\ell_p$-space, Wikipedia gives two alternatives for the $p\to 0$ limit of a $p$-norm. One is
$$\lVert x\rVert_0 = \sum_n 2^{-n}\frac{\vert x_n\rvert}{1+\lvert x_n\rvert}$$
and the ...
- 17.1k
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Proving the spectrum of an operator defined over a finite dimensional space is the set of eigenvalues.
I am trying to understand why the spectrum of an operator $T$ defined over $\mathbb{C}^N$ is precisely the set of eigenvalues.
My first thoughts are that for an eigenvalue, $\lambda,$ $(A - \lambda I)$...
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1
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49
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Let T be a bijective bounded linear operator on a Hilbert space H. Assume $T^{-1}$ exists and is bounded. Show that $(T^*)^{-1}$ exists.
This is my current attempt to this problem:
$T^*$ is injective:
I want to show $\mathcal{N}(T*) = \{0\}.$
Consider $T^*(u)=0.$ Then $<T^*(u), v> = 0$ for each $v \in H.$ By the definition of $T^*...
3
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1
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114
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Prove that $f(\| x+y \| ) = f(\| x \| ) + f(\| y \| )$ assuming that is a convex non-decreasing non-negative function and $y=kx$ ($k > 0$)
I'm trying to solve the following question:
Assume that $f : [0, \infty) \rightarrow \mathbb{R}$ is convex and continuously differentiable where $f(0)=0$ and $f'(0)=0$. $X$ is a normed vector space ...
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3
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Polynomials with only even terms are dense in the set of polynomials in $[0,1]$ with sup norm.
Let ${\cal P}$ set of all polynomials on one variable in $[0,1]$. It can be equipped with the supremum norm to obtain a normed space. Consider ${\cal Q}= \{a_0 + a_2x^2+a_4x^4+\dots a_{2k}x^{2k}: a_j\...
- 304
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301
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Defintion of distributions why not define with complex conjugate
For a complex valued locally integrable function $f$ on an open set $U \subset \mathbb{R}^{n}$, I saw many sources defined distribution induced by $f$ as $\phi \mapsto \int f\phi \, dx$.
If $\phi$ is ...
- 1,948
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On the continuity a function given by evaluating compact subsets of continuous functions
Let $B$ be a closed ball in $\mathbb{R}^n$ and write $C(B)$ for the Banach space (with respect to the supremum norm) of the continuous real-valued functions on $B$.
Now given a compact subset $K$ of $...
- 590
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36
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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}}$$
I was able to ...
- 9
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55
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Analytic continuation inherit invariance?
If a function $w_N$ on $(\mathbb{R}^{D})^{N-1}$ ($D$ and $N$ are integers) is invariant under some transformations, is its analytic continuation to a set in $(\mathbb{C}^{D})^{N-1}$ also invariant ...
4
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2
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59
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About BMO space on smooth bounded domains
Let $\Omega$ be any domain(open and connected) in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty ...
- 24.5k
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Prove $\|u\|_{C^{2,\alpha}(\mathbb{R}^3)}\leq C\|f\|_{C^{0,\alpha}(\mathbb{R}^3)}$ for $u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{f(y)}{|x - y|}dy$
Assume $ 0 < \alpha < 1 $, and the function $ f \in C^{0, \alpha}(\mathbb{R}^3) $ has compact support. Define the function
$$
u(x) = \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{f(y)}{|x - y|} \, dy....
- 759
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1
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75
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Are probability distributions the same as these distributions in mathematical analysis? (simple question)
Consider this definition of distributions that we see in analysis.
I have been brushing up on my probability and statistics knowledge. I already know that the term "distribution" has ...
- 1,330
1
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1
answer
62
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An easy proof of weak*-weak* continuity of the canonical mapping $\psi: X^*\to X^{***}$
I was trying to prove that the canonical mapping $\psi: X^*\to X^{***}$ is $\text{weak*-weak*}$ continuous. I have tried in the following way:
Let $\{x_\alpha^*\}$ be a net in $X^*$ such that $x_\...
- 560
0
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0
answers
56
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Convergence of $\cos(A)$ for operator $A$ in Banach space [closed]
For which linear bounded operators $A$ in Banach Space the following series converges in norm: $$\cos(A) = \lim_{n \to \infty}\sum_{k = 0}^{n}(-1)^k \frac{A^{2k}}{(2k)!}$$
I think it can be solved if ...
- 9
1
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0
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54
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$\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on $L^{p}$ if it is bounded on $L^{q}$
On wikipedia, it was mentioned that for $\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on $L^{p}$ if it is bounded on $L^{q}$. I wonder what proof of this would be.
For any $...
- 1,948
1
vote
2
answers
52
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Upper bound for integral operator from $L_2[0, 1] \to L_2[0, 1]$ [closed]
How can we bound the norm of integral operator $H$ in $L_2[0,1] \to L_2[0, 1]$?
$H=\int_0^tx(\tau)d\tau$
- 19
1
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0
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78
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do we have an estimate $\|\partial_v u\|_{H^{-1/2}} \leq \|\nabla u\|_{L^{2}(\partial\Omega)}$?
Let $u \in H^1(\Omega)$ such that $u$ is a solution for the problem $Lu = F$ in $\Omega$ and $u = g$ on $\partial\Omega$, where $g \in H^{1/2}(\partial\Omega)$ and $F \in L^2(\Omega)$. We assume that $...
- 303
0
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0
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55
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Let $f(x)\in L^p(\mathbb{R})$, $1\leq p<\infty$. Prove that $\lim_{|h|\to+\infty}\int_{\mathbb{R}}|f(x)+f(x-h)|^p dx=2\int_{\mathbb{R}}|f(x)|^p dx$ [closed]
Let $f(x) \in L^p(\mathbb{R})$, $1 \leq p<\infty$. Prove that
$$
\lim _{|h| \to+\infty} \int_{\mathbb{R}}|f(x)+f(x-h)|^p d x=2 \int_{\mathbb{R}}|f(x)|^p d x
$$
This is a question in my recent ...
0
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0
answers
32
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generator of a semigroup properties $A$ and $A-\alpha I$
at the moment I'm understanding a proof of a theorem about generating a C0-semigroup. Here's the theorem and the proof with associated definitions to understand it (see below).
My question is what ...
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
-1
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0
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14
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Can someone help me look into this proof of Measure Preserving transformation. How do i extend it to all the B_i in the Borel sigma algebra. [closed]
Let $(X, \mathcal{F}, \mu)$ be a measure space, where:
...
- 1
0
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2
answers
55
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Convergence of a sequence of function $t^n\sin(1-t)$
I should check if sequence of function $x_n(t)=t^n\sin(1-t)+t^2$ converge in $C[0,1]$. I find that if it converges, it has to be $x_n \to x(t)=t^2$.
So I have to prove $\max(t^n\sin(1-t) ) \to 0$, but ...
- 481
-1
votes
1
answer
28
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Computing the norm of bounded operator on the space of summable sequences using unit vector basis [closed]
Let $X$ be a normed space and $L$ a bounded linear operator from the space of summable sequences $\ell_1$ to $X$, and let $e_i$ be a unit vector sequence in $\ell_1$, i.e. a sequence with $1$ in the $...
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