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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Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
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32 views

Non-integer order derivative

I do not know much about fractional calculus, except what I have read in a few short posts at MSE and https://en.wikipedia.org/wiki/Fractional_calculus. I know that order of a derivative can be ...
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49 views

Mourre Adjoint: Bounded Maps (III)

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: ...
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What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
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37 views

Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$

Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq ...
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38 views

Exercise 8.1 in Brezis' Functional Analysis

Consider the function $$u(x) = \frac{1}{(1+x^2)^{\frac{\alpha}{2}}} \frac{1}{\ln(2+ x^2)} \qquad\; x\in \mathbb{R}$$ with $0<\alpha<1$. Check that $u\in W^{1,p}(\mathbb{R})$ for all $p\in ...
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19 views

$H_1\subset H_2$ Hilbert spaces with different norms. Which elements of the second can be expressed by an ONB of the first?

Suppose $H_1\hookrightarrow H_2$ are Hilbert-spaces with different scalar products and the inclusion is dense (e.g. $W^{1,2}\hookrightarrow L^2$). Suppose $(b_i)_{i\in\mathbb{N}}$ is an orthonormal ...
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30 views

A question on maximal monotone operators

Definition: Let $H$ a Hilbert space. An unbounded linear operator $A: D(A) \subseteq H \to H$ is said to be monotone if it satisfies $$\forall v \in D(A),\ (Av, v) \ge 0 $$ It is called ...
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41 views

$C(X)$ is separable when $X$ is compact?

$X$ is a compact metric space, $C(X)$ is separable when $X$ is compact where $C(X)$ denotes the space of continuous functions on $X$. How to prove it? And if $X$ is just a compact Hausdorff space, ...
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Infinitesimal generator is bounded

Consider a strongly continuous semigroup of bounded linear operators $S(t):X\to X$. The infinitesimal generator of $S(t)$ is the linear operator $A:D(A)\subseteq X \to X$ defined by ...
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Is there a sequence of continous function pointwise convergent to Riemann function?

I'm reading Baire's Category theory recently. One can find the following theorem in Chapter 4 of Stein's Functional Analysis: "Suppose that $\{f_n\}$ is a sequence of continuous complex-valued ...
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25 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
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34 views

Part (e) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (e) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
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24 views

Part (d) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (d) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
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1answer
25 views

Are Normed linear spaces $T_4$(Normal Hausdorff)?

Do normed linear spaces have the properties of a normal hausdorff space? I just sat an exam and I couldn't work out how to prove something initially, then I assumed that normed linear spaces are ...
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22 views

Can we do better than zero padding of FFT?

My background is in signal processing and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
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1answer
37 views

Part (c) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (c) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
5
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1answer
53 views

Part (a) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
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0answers
49 views

Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
0
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0answers
50 views

Inequality of Projections

Let $M$ be a $selfadjoint$ $subalgebra$ of $B(H)$, $x$ be a $positive$ $operator$ and $p$ be a $projection$ of $M$ such that $pxp\neq0$.Then show that there exists a non zero $projection$ $q$ of $M$ ...
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1answer
45 views

Show that linear finite rank operators are open mappings [duplicate]

Suppose $X$ and $Y$ are topological vector spaces, $dim(Y) < \infty$, $\Lambda : X \rightarrow Y$ is linear, and $\Lambda (X) = Y$. Prove that $\Lambda$ is an open mapping. Thanks in advance. ...
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31 views

Dimension of a subspace of a Hilbert space

Suppose $\{A_n\}$ is a sequence of operators on Hilbert space $H$ such that $\dim (\overline{\operatorname{Im} A_n})\leq \alpha$ where $\alpha\geq N_0$. If $A_n\to A$ uniformly, then $AH=\lim A_n ...
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1answer
38 views

Is a maximum function concave?

A function such as $$\max (X, Y)$$ given $X\geq 0$, $Y \geq 0$. Because It would form an inverse L shape on the graph, and that looks like a concave function. However my teacher said that the function ...
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40 views

Inner measure (inner set function) on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
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60 views

Is every projection on a Hilbert space orthogonal?

I'm highly doubtful that the answer is "yes," but I fail to see what's incorrect about this very basic proof I've thought of. If someone could point out my error, I'd appreciate it. My logic is as ...
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1answer
20 views

Double annihilator of subspace of dual space

If $X$ is a Banach space then it's quite straightforward to show that for $A$ a subspace we have $\bar{A} = {(A^{\circ})}_{\circ}$ and so if $A$ is finite dimensional then $A = {(A^{\circ})}_{\circ}$. ...
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21 views

space of continuous functions C(X,R): can one make it compact by making the domain X compact?

Consider a set of sequences $u(t)$ with discrete index $t=1,2,...$ where $u(t)\in[a,b]$, i.e. all values are taken from a finite interval. The space of such sequences is not compact. To make it ...
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1answer
22 views

Will this matrix have bounded orbit too?

Trying to solve: If $A \in GL(n,\mathbb{R})$ and $Spec(A)\cap \mathbb{S}^1 = \emptyset$, then $ \exists c > 0$ such that for all non-zero $x \in \mathbb{R}^n$ there is $m \in \mathbb{Z}$ ...
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32 views

A linear function $l$ on continuous space $C[0,1]$ is continuous?

Consider the space of continuous function on $[0,1]$, that is $C=C([0,1])$. Then any linear functional $l$ on $C$ is always continuous ? I think so, but no simple idea to proved.
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Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
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21 views

Why is $(-\Delta)^{\frac 12}$ a nonlocal operator (with spectral definition) but the ordinary Laplacian is not?

Take the Neumann Laplacian on a bounded domain $-\Delta$. We define $(-\Delta)^{\frac 12}u = \sum_{k}\lambda_k^{\frac 12}(u,w_j)_{L^2}w_j$ where $w_j$ and $\lambda_j$ are the eigenelements when $u$ ...
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14 views

C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
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17 views

Find the inverse of an operator, and determine is it bounded.

I've been doing some similar problems, but I got stuck on this one... and I have a feeling I'm running in circles trying to solve it. Any help appreciated! Problem: We have an operator: $$ A : ...
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14 views

Prove map is inflating

Let $T:X\to Y$ be a continuous linear open map between two Banach spaces. Prove that $\exists K\in\mathbb{R}$ such that for each $y\in T(X)$ we have $$T^{-1}(\{y\})\cap B_{K||y||}(0)\neq\varnothing$$ ...
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2answers
31 views

Compact Hausdorff spaces are normal

I want to show that compact Hausdroff spaces are normal. To be honest, I have just learned the definition of normal, and it is a past exam question, so I want to learn how to prove this: I believe ...
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2answers
36 views

Proving that a compact subset of a Hausdorff space is closed

I am having trouble understanding the answers here. I am trying to prove that a compact subset of a Hausdorff space is closed. Following the proof is difficult, perhaps because Brian reused letters ...
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2answers
104 views

The convex hull of every open set is open

Let $X$ be a topological vector space. Prove that the convex hull of every open subset of $X$ is open. I tried using definition of Convex Hull and Open Set, but I couldn't prove the statement.
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31 views

What is topological learning?

I am getting this term topological learning in few places for example a reference is below at section 1.1.2: http://virenjain.org/thesis/VirenJainThesis_official.pdf Can anyone point out what ...
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How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
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21 views

Second Order Mean Value Inequality In Banach Space

I have some confusions about proving the following theorem from Luenberger's Vector Space Optimization book, Proposition 2 p.176: $\textbf{Claim:}$ Let $X$ be a vector space and Y be a normed space. ...
2
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2answers
48 views

Can the Dirac Delta, $\delta $, be obtained by taking the limits of a rectangular pulse?

Can the Dirac Delta, $\delta $, be obtained by taking the limits of a rectangular pulse of width w and amplitude 1/w (ie. as w tends to zero)? How does the results differ from using the Gaussian, the ...
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2answers
34 views

Prove uniform convergence of $\sum_{e \in \xi} \langle Th, e \rangle e$ for $\| h \| \leq 1$

Let $\xi$ be a basis for Hilbert space $H$. From Parseval's Identity, for every $x \in H$ we have $x = \sum_{e \in \xi} \langle x, e \rangle e$. Thus, for every bounded operator $T : H \rightarrow H$ ...
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If $AT = TA$ for every continuous compact operator $T$, then $A$ is a multiple of identity

Given a Hilbert space $H$, let $A: H \rightarrow H$ be a bounded operator. Show that if $AT = TA$ for every continuous compact operator $T : H \rightarrow H$, then $A$ is a multiple of identity ...
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50 views

Show that every continuous finite rank operator $T$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$

Can someone help me with this question? Suppose that $H$ and $K$ are Hilbert spaces. Show that every operator $T \in B_{00}(H, K)$ can be written as $\sum_{i=1}^n \lambda_i x_i \otimes y_i$, where ...
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3answers
141 views

Show that $f \overset{T}{\rightarrow} \frac{1}{x} \int_0^x f(t) dt$ is Bounded, and is NOT Compact in $L^2(0, \infty)$

Can someone help me with this question? Let $f \in X = L^2(0, \infty)$, and define \begin{equation} (Tf)(x) = \frac{1}{x} \int_0^x f(t) dt \ . \end{equation} Show that $T$ is Bounded, and is NOT ...
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1answer
62 views

Show that $\{ x_n \} \overset{T}{\mapsto} \{ \sum_{k=1}^{\infty} a_{nk} x_k \}$ is compact

Can someone help me with this question? Let $\ell^2$ be the space of complex sequences $\{ x_1, x_2, \ldots \}$ that $\sum_{n=1}^{\infty} \lvert x_n \rvert ^2 < \infty$. If $\mu$ be Counting ...
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1answer
22 views

Adjoint of canonical expansion of compact operator

Lets say I have given a rank-$n$ operator $A = \sum^n_{k=1} \lambda_k \langle u_k, \cdot \rangle v_k$. Then it is straightforward to compute its adjoint as $A^\ast = \sum^n_{k=1} \lambda_k \langle ...
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1answer
20 views

Reference for denseness of testfunctions in sobolevspace

for my thesis I need a reference for a proof that $C_0^\infty(\mathbb R)$ is dense in $W^{2,2}(\mathbb R)$ in respect to the Sobolev-$\| \cdot \|_{W^{2,2}}$-Norm. I have tried Google but I can't find ...
2
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26 views

Compatibility of operator spaces and tensor product norm

I have a problem with understanding the notion of complete boundedness in tensor notation. One possible way of saying a mapping $\phi\colon \mathcal{A}\to \mathcal{B}$ is completely bounded is to ...
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1answer
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Why does $\mathop {\lim }\limits_{n \to \infty } \left\| {T{x_n} - \lambda {x_n}} \right\| = 0$? [duplicate]

Let $T \in B(X)$ and $\lambda \in \sigma (T)$, why is there a sequence like $\left\{ {{x_n}} \right\}$ in $X$ with $\left\| {{x_n}} \right\| = 1$ such that $\mathop {\lim }\limits_{n \to \infty } ...