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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).

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40 views

On injective, bijective, surjective

Consider a function $f: \mathcal{A}\rightarrow \mathbb{R}$ where $\mathcal{A}\equiv \{a_1,a_2,a_3,a_4\}\subset \mathbb{R}$ . We know that $f$ is injective, i.e., $f(a_j)\neq f(a_k)$ $\forall a_j\neq ...
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2answers
44 views

What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
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2answers
49 views

Functional analysis: locally compact hausdorff space

Let $X$ be a locally compact Hausdorff space. Show that every function in $C_0(X)$ (continuous functions that vanish at infinity) can be arbitrarily uniformly approximated by functions in $C_{00}(X)$ (...
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0answers
23 views

completeness of Laguerre polynomials

Can you help me to prove that system of Laguerre polynomials $$ L_n = \dfrac{e^t}{n!}\dfrac{d^n}{dt^n} (t^n e^{-t})$$ is completeness in space $L_2((0, \infty),e^{-t}dt)$ ? i have idea of proof: ...
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2answers
48 views

Interior point in convex set on n.v.s

I'm reading Peter Lax's Functional analysis, and I have a question about a definition: Definition. $X$ is a linear space over the reals, $S$ a subset of $X$. A point $x_0$ is called an interior ...
2
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1answer
94 views

Compact operators and weak convergence

Let $X$ and $Y$ be Banach spaces. (a) Let $T \in \mathcal{L}(X, Y )$. For each sequence $(x_n)_{n \geq 1}$ in $X$ and each $x \in X$, show that $x_n →x$ weakly, as $n \rightarrow \infty$ ,implies ...
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2answers
71 views

How can sequences satisfy vector space properties? If sequence is not a vector.

How can sequences satisfy vector space properties? If sequence is not a vector. An example semantic problem occurs, when one needs to find a zero element or zero $\bar{0}$. In order to do this for ...
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1answer
43 views

Asymptotic behaviour of gradient flows for $t \to \infty$

I have often heard about the asymptotics of gradient flows converging to some "equilibrium point" as $t \to \infty$. This concept has come to my ear by word of mouth multiple times and is often ...
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0answers
15 views

on completeness of a complete locally convex space w.r.t. the weak topology

Let $E$ be a complete locally convex space, let $E'$ denote its topological dual, and let $\sigma(E,E')$ denote the weak topology on $E$. Is it true that the space $E$ is complete when equipped with ...
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1answer
29 views

Continuity implies boundness in Frechet Spaces?

Consider two Frechet spaces $(X,d_X),\;(Y,d_Y)$. Let $T:X \rightarrow Y$ be a continues linear operator. Is it true that $T$ is bounded(or Lipschitz) in the following sense: $$\exists \alpha > 0 \; ...
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1answer
24 views

Alternative form for Liapunov inequality

Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then $$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$ My teacher ...
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1answer
34 views

A book for questions on functions

I recently learnt functional analysis and I'm pretty comfortable at solving its questions but when it comes to more complex statements(though it may not be complex but for me they seem they are)I get ...
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0answers
13 views

Integral Convergence Estimate with cut-off function

Let $u \in L^{\infty}(\Omega)\cap H_{0}^{1}(\Omega)$ and define a cut-off function $\eta_{R} \in C_{0}^{\infty}(\mathbb{R})$ for $\Omega \subset \mathbb{R}$ an unbounded (interval) domain as follows $...
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1answer
31 views

Property of convex sets.

Let $X$ be a normed space, $S$ a subset of $X$ and let $x_0\in X$. Consider the two propositions: (1) There exists $r>0$ such that $B(x_0,r)\subset S$. (That is, $x_0$ is an interior point of $S$.)...
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31 views

Show that for $|a|<1$ the series $\sum_{n=1}^{\infty}a^nf_n$ converges in $L_2([-1,1])$.

Let $f_n(x)=x^{n+1/2}$ for $n\geq 1$ and $x\in [-1,1]$. Show that for $|a|<1$ the series $\sum_{n=1}^{\infty}a^nf_n$ converges in $L_2([-1,1])$. First, I have shown that $\sum_{n=1}^{\infty}a^nf_n(...
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31 views

Proof $f\in N(A^{*})^{\perp}$

I'm supposed to show $f\in N(A^{*})^{\perp}$ in a proof. Here, we're dealing with a compact linear injective operator $A:X\rightarrow Y$, where both $X$ and $Y$ are Hilbert spaces. Furthermore, $A$ ...
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2answers
65 views

Is a Hahn-Banach extension always continuous?

We proved the following version of the Hahn-Banach extension theorem in a course I'm taking: Theorem (Hahn-Banach): Let $X$ be a real vector space and $q : X \to \mathbb{R}$ be sublinear. Let $U \...
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3answers
52 views

Prove that $ \|x\|=\sup\{|f(x)|:f\in X^*, \,\|f\|=1\},$ where $x\in X$ and $X^*$ denotes the dual space of $X$.

Let $x$ be an element of a normed linear space $X$ and let $X^*$ denote the dual space of $X$. Prove that \begin{align} \|x\|=\sup\{|f(x)|:f\in X^*, \,\|f\|=1\} \end{align} MY TRIAL It suffices to ...
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1answer
46 views

variational formulation of second order differential equation

I am given the following differential equation. Let $\Omega = (a,b)\subset\mathbb{R},\ f:\Omega \rightarrow\mathbb{R},\ \alpha,\beta \in \mathbb{R}$ and $$ -u'' + u = f \\ u(a)= \alpha, u(b) = \beta ...
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0answers
33 views

Nonlinear approximation by piecewise constants and the expectation of the error

Suppose $\Omega=[0,1]$ and $f$ is continuous, monotonic, and of bounded variation on $\Omega$ with $M:= \text{Var}_\Omega(f).$ Let $\| \cdot\|:=\| \cdot\|_{L_\infty(\Omega)}.$ Let $T=\{0=t_0,...,t_n=...
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2answers
57 views

Expectation of $XY$ bounded for all bounded $Y$ implies $X$ is $L^p$

I'm trying to prove: Let $X$ be a real random variable, $p, q \in (1,\infty)$, $\frac 1 p + \frac 1 q = 1$. If there is $C < \infty$ such that $|\mathbb E[XY]| \leq C ||Y||_q$ for any bounded ...
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2answers
97 views

Prove that if $f(x)=f(y)$ for all $f\in X^{*},$ then $x=y$

Can you check if my proof is correct? Let $X$ be a normed linear space. Prove that if $f(x)=f(y)$ for all $f\in X^{*},$ then $x=y$ Let $f\in X^{*}$, then $f$ is a bounded linear functional. Assume ...
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0answers
25 views

Brezis excercise 4.12: $L_p$ is uniformly convex for $1<p\leq 2$

Let $1<p<\infty$. Prove that there is a constant $C$(depending only on $p$) such that $$|a-b|^p\leq C(|a|^p+|b|^p)^{1-s}(|a|^p+|b|^p-2|\dfrac{a+b}{2}|^p)^s$$ for all $a,b\in \mathbb{R}$ and $s=\...
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0answers
39 views

Approximation of a two-variable function by tensor products

Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function. We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...
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1answer
50 views

uniform boundness principle (Banach- Steinhaus)

I am a beginner of functional analysis and I can't understand at all the Banach Steinhaus theorem: Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily ...
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0answers
24 views

Isometric isomorphism between $L^2$ and $\mathcal{L}^2$

I was reading and trying to understand the proof that the space $\mathcal{L}^2 (\mathcal{H})$ (Hilbert-Schmidt operators) is made by all the $T_K:L^2(X,\mu) \rightarrow L^2(X,\mu)$ with $K \in L^2(X \...
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0answers
79 views

A standard example of calculus of variation that I don't understand

Let us consider the problem $$ \min \left\{ \int_0^1 \psi\left(\dot{u}\right) \ dx: u \in \mathcal{C}^1([0,1]), u(0) =0, u(1)=2 \right\} $$ where $\psi \colon \mathbb{R} \longrightarrow \mathbb{R}$...
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1answer
15 views

Regarding continuous function not in the Disc algebra

Let $D=\{z\in\mathbb{C}: |z|<1\}$. $C(\bar{D})=\{f:\bar D\longrightarrow \mathbb{C}: f \;\text{is continuous on}\; \bar{D}\}$ $A(D)=\{f\in C(\bar{D}): f \;\text{is analytic in} \;D\}$ Can you ...
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1answer
31 views

Theorem 3.18, Rudin's functional analysis

Just a quick question about the the following theorem In a locally convex space $X$, every weakly bounded set is originally bounded and viceversa. Proof: Since every weak neighborhood of $0$ ...
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1answer
36 views

Banach-Alaoglu theorem, Rudin's functional analysis.

Few questions about the theorem If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$ K = \left\{\lambda \in X^* : |\Lambda x | \leq 1 \; \text{for every} \; x \in V \right\}...
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1answer
32 views

Taylor expansion for Gâteaux derivative

Let $\mathbb{X}$ be a normed Space and $ f: \mathbb{X}\mapsto\mathbb{R} $ is twice Gâteaux differentiable (not necessary Fréchet differentiable). Is it possible to build a Taylorexpansion for $f$ in ...
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1answer
36 views

$F:A\rightarrow B$ be compact operator. Show closure subset of $B$ contains $0$.

$A,B$ are Banach spaces, $A$ infinite dimensional and $F:A\rightarrow B$ a compact linear operator. Show that the closure of $\{Fx:||x||=1\}\subset B$ contains $0$. I've managed in a previous ...
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1answer
17 views

A subadditive functional which is not a convex functional

I have been thinking about this for a while now, is there a subadditive functional which is not a convex functional?
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2answers
26 views

Rudin's functional analysis theorem 3.12

Suppose $E$ is a convex subset of a locally convex space $X$. Then the weak closure $\overline{E}_w$ of $E$ is equal to its original closure $\overline{E}$. The proof starts as follows $\overline{...
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1answer
35 views

Continuous functions vanishing at infinity on a non-locally-compact space

Let $X$ be a topological space which is not locally-compact (e.g., an infinite-dimensional Hilbert space). Let $C_{0}(X)$ denote the space of complex-valued, continuous functions vanishing at infinity ...
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1answer
13 views

Question on a proposition of uniorm convergence of function series

The Proposition states if $(f_n)_{n\in\mathbb{N}}$ is a secuence of functions $f_n:D\rightarrow \mathbb{C}$ with $\sum_{n=1}^{\infty} ||f_n||_D < +\infty$ then the series $\sum_{n=1}^{\infty}f_n$ ...
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0answers
41 views

Is this seminormed space (intersection of Banach spaces) non-empty?

In this paper of Zubelevich the author consider a a scale of Banach spaces $\{(E_{s},\|\cdot\|_{s}):0<s<1\}$, that is to say, each $(E_{s},\|\cdot\|_{s})$ is a Banach space and $$ E_{s+\delta}\...
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1answer
43 views

Operator norm of $T:l^{2}\rightarrow l^{1}$ where $Tx=(x_{1},x_{2}/2,x_{3}/3,x_{4}/4,…)$

As the title states, I need to compute the operator norm of a linear operator $T:l^{2}\rightarrow l^{1}$, where $$Tx=\left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},\frac{x_{4}}{4},... \right)$$ Using ...
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1answer
45 views

Show that $m\in M$ is unique

Let $S$ be a subspace of a normed linear space, $X$ and $x_0\in X\backslash S$. Consider the subspace spanned by $M,$ i.e. \begin{align} M:=[S\cup \{x_0\}]=\{m=x+\alpha\,x_0:\,x\in S,\;\text{for some}\...
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1answer
21 views

Banach Space,pde,compact operator

$X, Y, Z$ are Banach Space, $f: X \to Y$, $g: Y \to Z$ are bounded linear operators. Show: if $f$ or $g$ is a compact operator , then $g \circ f: X \to Z$ is a compact operator.
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1answer
48 views

a reference for the isomorphism $\ell_1(E')\cong [c_0(E)]'$

Let $E$ be a Banach space, and let $E'$ denote its topological dual. Let us consider the spaces $\ell_1(E')$ and $c_0(E)$ defined by $\ell_1(E')=\{(x_n^{'})_{n=1}^\infty\subset E': \sum_{n=1}^\...
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1answer
50 views

Excercise 4.3 (3) in Brezis. Convergence in $L_p$

Let $(f_n)$ in $L_p(\Omega)$ $1\leq p< \infty$ and $(g_n)$ bounded in $L_{\infty}(\Omega)$ assume that $f_n \rightarrow f$ in $L_p(\Omega)$ and $g_n \rightarrow g$ a.e. Prove that $$f_ng_n\...
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0answers
19 views

Reference Request - Distributions

Is there any reference on distributions that proceeds in a motivational way from the Heaviside function to Schwartz's work.
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1answer
39 views

Spectral Theorem for Unitary Operator

It is well known that the following - in many literature - called the Spectral Theorem for Unitary Operator. I would like to know where i can find further information about it and its proof.
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0answers
27 views

Rudin's functional analysis theorem 3.10, proof that multiplication is continuous

Suppose $X$ is a vector space and $X'$ is a separating vector space of linear functionals on $X$. Then the $X'$-topology $\tau'$ makes $X$ into a locally convex space whose dual space is $X'$. ...
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1answer
43 views

Spectrum of operator A

Let $H$ be a Hilbert space and $A \in B(H)$. How to prove that if $|\lambda| = \|A\|$ and $\lambda \in W(A)$, then $\lambda \in \sigma_p(A)$? Here $W(A)$ is the numerical range of the operator $A$: $$...
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2answers
25 views

Rudin's functional analysis, section 3.8 (c) proof.

From Rudin's functional analysis. If $X$ is a compact topological space and if some sequence $\left\{ f_n \right\}$ of continuous real-valued functions separates points on $X$, then $X$ is ...
2
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1answer
41 views

Proving that every finite-dimensional normed space has the same norm as the euclidean n-dimensional space

I have a problem where I have to show that every $n$-dimensional normed space $E$ has the same norm as the euclidean space $E_n$. Here's what I've got: Since $E$ is $n$-dimensional then for the ...
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2answers
34 views

How to prove that the Haar system is orthonormal?

Haar wavelets are defined as: $$ \psi_{0,0}(t) = \begin{cases} 1, \text{ for } 0<t< 1/2\\ -1, \text{ for } 1/2<t<1 \\ 0, \text{ otherwise } \end{cases} $$ Where mother wavelet is$$\psi_{...
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0answers
20 views

The difference between the nonlocal and local conditions problems

In some of Boundary value problems involving ordinary differential equations,, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed. In this paper: ...