Functional analysis is 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, as well as measure, ...

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Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded?

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded? I don't even know where to start proving or disproving, ...
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71 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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60 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
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34 views

Calculus Problem___Prove $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ for any $t>0$

Given $g \rightarrow R$ continuous and bounded, let $$u(x,t)=\frac{x}{\sqrt{4 \pi}}\int_{0}^t \frac{1}{(t-s)^{3/2}}e^{\frac{-x^2}{4(t-s)}}g(s)ds$$. Prove that $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ ...
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52 views

Schwartz Space is closed under differentiation and multiplication by polynomials.

Schwartz space is closed under differentiation and multiplication by polynomials. In addition, if $f$ is a smooth function will all derivative bounded and $\psi$ is a Schwartz function, then ...
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36 views

Is the uniform limit of continuous functions continuous for topological spaces?

Question: [For notations see the context given below.] If $E$ is a topological space and $(f_n)_{n\geq1}$ is a Cauchy sequence in $\mathcal{C}\mathcal{B}(E)$, then we know $(f_n)_{n\geq1}$ ...
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47 views

Understanding the Proof of the Arzela-Ascoli Theorem from Carothers

The below is the proof for the Arzela-Ascoli Theorem from Carothers' Real Analysis. I had a few questions regarding some steps in his proof which I have put in blue. If anyone could explain the blue ...
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51 views

Invertible, Positive and Isometry Operator.

Let $T ∈ L(V )$ and $T = SP$, where $S$ is an isometry and $P$ is a positive operator. Prove that $T$ is invertible if and only if $P$ is invertible. Here is my approach: $\implies:$ $T = SP$ by ...
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28 views

$Y$ is a Banach space if $B(X,Y)$ is a Banach space

Let $X$ and $Y$ be normed linear spaces, $X\ne\{0\}$. Let $B(X,Y)$ be the collection of all bounded linear operators from $X$ into $Y$ with the operator norm. Suppose that $B(X,Y)$ is a Banach space. ...
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23 views

Constant for polynomials of limited degree

I'm trying to prove the following: There exists a constant $C \in \mathbb{R}$ such that for all polynomials $f(t) \in \mathbb{R}[t]$ of degree not greater than $ 2014 $ we have $$ |f(10)| \leq ...
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43 views

$A$ is a symmetric operator ? Please criticize my proof.

Let $A:L^2([0,1])\to L^2([0,1])$ given by $$ Af(t)=\int_0^1K(s,t)f(s)ds, $$ where $K$ is a mensurable square integrable operator, i.e $\int_0^1\int_0^1|K(s,t)|^2\,dsdt<\infty$. $A$ is acompact ...
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38 views

$C_b(X)$ is non-separable for $X$ non-compact

If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$. I was trying like this, but got stuck at middle: Take a ...
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34 views

Energy dissipation

I've been asked to prove the following, but I don't find the way.. Let $\Omega\subset\left\{0<x_n<a\right\}$ be a subset of $\mathbb{R}^n$ such that it is bounded in the $n^{th}$ coordinate. ...
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44 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
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47 views

Why is $\widehat{a}$ bijective?

I am trying to understand the proof of the following theorem: Let $A$ be a unital Banach algebra generated by $1$ and $a$. Then $A$ is abelian and the map $\widehat{a}:\Omega(A) \to \sigma (a), \tau ...
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31 views

Closed Operators: Empty Spectrum

Are there operators on Hilbert space having empty spectrum? (Surely, for Banach spaces they do exists.) Necessarily, they must be closed and can't be normal.
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38 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
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94 views

Almost everywhere convergence of a bounded sequence in $H_0^1(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement (e.g. $\Omega = \mathbb{R}^N \setminus \overline{B(0;1)}$), and $(u_n)$ be a bounded sequence in $H^1_0(\Omega) = ...
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31 views

Confusion about the definition of self adjoint and formally self-adjoint

I have some confusion about the definition of self-adjoint operators and formally self-adjoint operators. Let me write down the background information. Let $H$ be a infinite dimensional complex ...
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19 views

When can we say that $T\in B(H)$ is $YS$ for some fixed $S$ in $B(H)$?

Let $T,S\in B(H)$ where $H$ is a Hilbert space. Suppose that for all $x\in H$, $\|Tx\|\leq \|Sx\|$. Could we then say that $T=YS$ for some $Y\in B(H)$? Would it help if $S$ were a contraction or ...
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40 views

Dense convex proper subset of the Hilbert space $l_2$: $\{x|\sum x_i=0\}$ [duplicate]

Let's consider the space $l_2$ (all sequences $x$ with $\sum x_i^2 < +\infty$) and its subset $Z = \{x|\sum x_i = 0\}$. I want to prove that the closure of $Z$ is $l_2$, but I can't. I tried to ...
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32 views

Is the algebraic interior relatively open in a closed convex set?

I am struggling with proving or disproving the following: Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall ...
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14 views

$L^{2}$ convergent, subsequence, directed family of points

I have a question about a convergence. Let $(E,\mathcal{B},m)$ be a measure space. I think the following assertion is very famous: Let $f_{n},f \in L^{2}(E;m)\quad(n=1,2,\cdots)$. If $f_{n}\to f $ ...
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30 views

Does this funcion define a norm on $\mathbb{C}^n$?

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} ...
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45 views

Prove an integral operator is compact

The statement is like this, $K\subset\mathbb{R}$ is compact, the operator $A:L^\infty(K)\mapsto L^\infty(K)$ is defined by $f(x)\mapsto\int_K k(x,y)f(y)dy$. For $x\neq y$, $|k(x,y)|\leq ...
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23 views

$L^{2}$ convergence, pointwise convergence

I have a question about $L^{2}$ convergence, pointwise convergence. Let $(E,\mathcal{B},m)$ be a measurable space and $\mathcal{D}$ be a dense subset of $L^{2}(E;m)$. The following assertion is ...
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66 views

Why is this operator self-adoint

We have that $\lambda, \overline{\lambda} \in \rho(T)$ and $\lambda \in \mathbb{C}$. Now, I want to show that a symmetric operator and closed operator $T: \operatorname{dom(T)} \rightarrow H$ must be ...
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45 views

Hilbert spaces of holomorphic functions

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps ...
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25 views

Dense subset in $l^{p}$

Let $p>1$ and $l^{p}=\{\{x_n\}_{n=0}^{\infty}:\sum_{n=0}^{\infty}|x_n|^p<\infty,x_n \in \mathbb{C}\}$. Then subset $Y=\{\{x_n\}_{n=0}^{\infty}:\sum_{n=0}^{\infty}x_n=0,\{x_n\}_{n=0}^{\infty} ...
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43 views

Show that the set of uniformly Lipschitz functions vanishing at $0$ is compact in $C[0,1]$

The question is: For $K$ and $\alpha$ fixed, show that $\{f\in \operatorname{Lip}_k \alpha : f(0) = 0\}$ is a compact subset of $C[0,1]$. I was going to attempt this by using by Arzela-Ascoli theorem ...
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32 views

A lemma about quasicentral-approximate-unit

Here is a lemma about quasicentral-approximate-unit: Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that ...
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71 views

Resolvent also self-adjoint operator

If I have a self-adjoint operator $U : \operatorname{dom}(U) \subset H \rightarrow H$ and $\lambda \in \rho(U)$, then I assume assume that it is correct that the operator $(U - \lambda I)^{-1} \in ...
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42 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
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33 views

$X$ is the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$, and $M=\{f\in X:f(0)=0\}$, show that $M$ is not closed.

Here is my question: Let $X$ be the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$. Let $$M=\{f\in X:f(0)=0\}$$ Show that $M$ is not closed. Show that the “quotient norm" ...
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36 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

$H$ is a Hilbert space, $M$ is a self-adjoint bounded linear operator on $H$ with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all $n$, both $S_n$ and $S_n - ...
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23 views

Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
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41 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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53 views

The topology on $C^\infty_c(\mathbb{R}^d)$ used for “distributions of compact support”

On the one hand, Eskin's book on PDEs tells me that I should be content to think of this topology as one "described" (not fully, and it's not even clear it's a topology) by the convergence of ...
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41 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
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How to create distribution function from sketch?

I'm playing with image manipulation based on various mathematical algorighms (such as edge detection). I'm also changing the colors in various ways just to see what comes out of it. Regarding this, ...
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17 views

Show that the matrix $(a_{j,k})_{j,k\in \mathbb{N}}$ induces a bounded operator on $\ell^2$.

I have a matrix $(a_{j,k})_{j,k\in\mathbb{N}}$ given by: $ a_{j,k} = \dfrac{1 -e^{-jk}}{jk + 1}$ and I need to show that this induces a bounded operator on $\ell^2$. I'm pretty sure Schur's test is ...
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34 views

canonical form of finite rank operators

Let $X,Y$ be banach spaces and let $T:X\rightarrow Y$ be an linear continuous operator with finite dimensional image $Im(T)\subset Y$. Now I want to prove that there exists continuous linear ...
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48 views

Showing an operator is essentially self-adjoint

I have a question about checking if an operator is essentially self-adjoint. Given the operator $$H=-\frac{1}{2}\partial^2_{r}-\frac{1}{r}\partial_r$$ with domain $C^{\infty}_0((0,\infty))$ (i.e. ...
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25 views

Discontinuous mapping between function spaces

Let $ C([0,1]) $ be a space of continuous real-valued functions over interval $[0,1]$ and $ \|f\|_2 = (\int_0^1|f|^2 \, dx)^{1/2} $ define a norm over this space. Prove that the following mapping: $ ...
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39 views

Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
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56 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
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63 views

what does it means $∥T(1_{[a,b]})∥$

For each $f\in L_{2}(0,\infty)$, we set $Tf:(0,+∞)\to \mathbb{C}$ with $Tf(s)=\frac{1}{s}\int\limits_{(0,s)}f(t)dt$. For each $0<a<b$ i want to show that $∥T(1_{[a,b]})∥_{2}\geq\frac{b-a}{\sqrt ...
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58 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
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34 views

In a Hilbert Space, if $\langle x,x_n \rangle \to 0$ then $\sup\{\|x_n\|:n=1,2,3,…\}<\infty$

Let $\mathbb{H}$ be a Hilbert space. Let $\{x_n\}$ be a sequence in $\mathbb{H}$ with the property that $\langle x,x_n \rangle\to 0$ as $n\to\infty$ for $x\in\mathbb{H}$. Show that ...
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59 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...