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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Existence of certain idempotents

Suppose $T$ is an idempotent (that is $T^2=T$) of infinite rank and co-rank on a separable Hilbert space. Can we find an idempotent $S$ such that $\overline{TS(H)}=(Id-S)(H)$?
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Spectral Measures: Scale Spaces (I)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote measurable calculus: ...
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Wot - open set in $B(H)$

To prove an exercise, I need to show if sot - subbase $N(0,h,r) = \{x\in B(H);~~\|xh\|<r\}$, for $h\in H$, is wot - open, then every sot - subbase is wot - open. Definition: $U$ is wot - open if ...
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15 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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2answers
38 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
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24 views

series of linear operators

Let $\mathcal{B}(\mathcal{H})$ be the Banach space of bounded linear operators on a complex, separable, infinite-dimensional Hilbert space $\mathcal{H}$. It is well known that ...
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1answer
14 views

How is completeness used in the Eberlein–Šmulian theorem?

So for weakly closed subsets of Banach spaces compactness and sequential compactness coincide, but upon studying the proof I can't put my finger on what exactly would go wrong if we dropped the ...
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1answer
20 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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38 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
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1answer
24 views

Show that Quotient of a C*-algebra is a C*-algebra in its own right

Let $A$ be a commutative C*-algebra over $\mathbb{C}$ and let $J$ be an ideal of $A$ such that $J\in Closed(A)$ and that contains $x^*$ if it contains $x$, for all $x\in A$. I know that $A/J$ is also ...
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14 views

Base for topological algebra

How to define the basis for topological algebra? what is finitely generated locally convex topological algebra?
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32 views

Sot convergence of a sequence of operators implies uniform convergence

Let $H$ be a Hilbert Space. Let $\{A_n\}$ be a sequence of bounded operators in $H$, and $A\in B(H)$. If $\|A_nf - Af\|\to 0$ uniformly for $f\in H_{\|.\|=1}\ $, prove that $\|A_n - A\|\to 0$. ...
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1answer
22 views

How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?

If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
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31 views

Is there a relation between vectors on these two spaces?

I've been reading lately one paper on Physics, which basically presents one gauge theory approach to the problem of swimming at low Reynolds number. I've been trying lately to rewrite some of the ...
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43 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
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46 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
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21 views

$L^1 ([0,1])$, bouned linear functional, absolute continuous function

I am studying for an Analysis prelim and was wondering if someone could perhaps either validate or invalidate my proof for the following problem: "Let $L^1 ([0,1])$ be the space of Lebesgue ...
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34 views

If $u \in L^2(0,T;L^2(\Omega))$ is $\int_{\Omega}\int_0^T |u(t,x)|^2$ defined?

Let $u \in L^2(0,T;L^2(\Omega))$ on some domain $\Omega$. We know that $$\int_0^T \int_{\Omega}|u(t,x)|^2$$ is defined, but is it equal to $$\int_{\Omega}\int_0^T |u(t,x)|^2?$$ Can I interchange the ...
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33 views

Range of normal operator and its adjoint are equal

On Wikipedia it is written that bounded normal operator in Hilbert space has the same range and kernel as its adjoint. I've been able to show equality of kernels and closures of ranges: ...
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46 views

how shall i'll prove if c={(x_n) :exists lim x_n}is a Hyperplane, dense, or closed? [on hold]

Let $$c=\{(x_n) :\exists ~ \lim x_n\},$$ where $c$ is included in $\ell^\infty$. How can I find a function $T$ such that $\ker(T)=c$? Also, after that, how can I see if $T$ is continuous? Thanks.
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2answers
57 views

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent?
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AAK theorem for finite dimensional Hankel matrix

Does the AAK theorem hold for finite dimensional Hankel matrix? Or maybe similar analysis exists? (From a quick look of the proof, it seems like the AAK solution has to be infinite dimensional ...
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20 views

Integral kernel of operators from the functional calculus theorem

Let $T$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ and take any bounded Borel function $F:[0,\infty)\rightarrow\mathbb{C}$. Does $F(T)$ -defined by the functional calculus ...
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16 views

Why's Daugavet equation important?

I've been recently studing Daugavet equation in $L^1[0,1]$ and $C[0,1]$. I understand most of the results I've found but I can't figure out why is it important to find operators that hold Daugavets ...
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25 views

is this conclusion true or false?

Let $\mathcal{A}$ be a factor Von Neumann algebra and $\Phi$ is a map on $\mathcal{A}$ which is injective and surjective and $\Phi(0)=0$. If $A, B, C \in \mathcal{A}$ and ...
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1answer
53 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...
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28 views

The Lax-Milgram Theorem for Banach Spaces

I wish ask by Question. A significaive (this is for Banach and not Hilbert space, for example a $L_p$ space with $p\neq 2$ or another as you see, a real Banach space) and understable (constructive ...
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2answers
33 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
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47 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
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1answer
31 views

Is $L^{1}(\Omega,\mu)$ only an algebra when $\Omega$ is a group?

Let $G$ be a locally compact group. Then we may snap our fingers, mention some measure theory results, and the group algebra $L^{1}(G)$ instantly appears. Is there an example of the more generalized ...
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1answer
34 views

How do I prove the converse of Stone-Weierstrass theorem?

Let $X$ be a locally compact Hausdorff space. Let $\bar \rho$ be the uniform metric on $\mathbb{R}^X$ and $\mathscr{A}$ be an $\mathbb{R}$-subalgebra of $C_0(X,\mathbb{R})$ which is dense in ...
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1answer
59 views

A condition for surjectivity of a linear map

Let $V,W$ be vector spaces (not necessarily finite dimensional!), and let $W^*$ the dual of $W$. Let $$A:V\longrightarrow W^*$$ be a linear map. What conditions do I have to put on $V$ and especially ...
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25 views

is intersection of a countable collection of dense, open subsets of a complete metric space also dense in X? [duplicate]

i do not know what this site is expecting to write. i've written my question above . saw in a question paper. again writing it. is intersection of a countable collection of dense, open subsets of a ...
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28 views

Dual of finite dimensional Hilbert space.

The dual space H* is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by $$\|\varphi\| = \sup_{\|x\|=1, x\in H} |\varphi(x)|.$$ The ...
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40 views

Definition of resolvent set

I'm having trouble understanding some subtlety of definition of resolvent set for given bounded operator A everywhere defined on some Hilbert space. Book I use (and many other sources) give the ...
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35 views

Question about weak derivatives

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$ We often say that $v$ is the ...
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1answer
132 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
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3answers
69 views

An equality in Hilbert spaces

To understand a proof in functional analysis I need to understand why the following equation is true: $$\lVert x\rVert^2 - \sum_{j=1}^n |x_i|^2 = \Biggl\lVert x-\sum_{i=1}^nx_ie_i\Biggr\rVert^2$$ ...
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1answer
34 views

Limit (in probability) of sequence of independent random variables

We have $\{X_n\}$ independent random variables which converge to $X$ in probability. I was asked to prove that $X$ is constant. My approach is to try to show that$Var(X)=0 \implies X$ constant, but i ...
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1answer
18 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
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1answer
35 views

What is a linear functional on continuous functions on the real line not given by a measure?

What is a positive linear functional on continuous functions on the real line not given by integration against a measure? I know that the dual of $C_c(\mathbb R)$ is the set of Radon measures, ...
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1answer
18 views

Reference for the statement “bilinear form $a$ is symmetric if and only if the operator $S$ is self-adjoint”

Thanks to Riesz representation theorem, a continues bilinear (sesquilinear) form on Hilbert space $$a: \mathcal H\times \mathcal H\rightarrow\mathbb R \ \ (\text{or} \ \ \mathbb C)$$ can be ...
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68 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
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How to construct a fundamental solutions of a PDE from well-posedness?

A fundamental solution of a linear operator $P$ on a manifold $M$ is a distribution $G$ such that: $$P(G)=\delta(x-y)$$ In formal terms this is stated as given a test function $\phi$ then: ...
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17 views

Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
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1answer
16 views

Question about affine isometric action

Recently, I read the book Kazhdan's Property (T). There is a lemma on the page 75 (Lemma 2.2.1) as following: Lemma. Let $\pi$ be an orthogonal representation of $G$ on $H^0$. For a mapping $\alpha: ...
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1answer
39 views

Is $H^2(\Omega)\cap H_0^1(\Omega)$ compactly embedded on $H_0^1(\Omega)$?

Considering $\Omega$ bounded and $\partial \Omega$ smooth. I already know that $H^2(\Omega)\cap H_0^1(\Omega)$ is continuously embedded on $H_0^1(\Omega)$, thus if I take a bounded sequence in ...
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1answer
55 views

Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
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44 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
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25 views

Prove or disprove that $φ_v:u\mapsto \langle\mathcal A u,v\rangle$ is in $V^*$

Let us consider a linear and continuous operator on a Hilbert space $V$, $\mathcal A:V\rightarrow V$, such that: $$\|\mathcal A u\|\leq M \|u\|, \ \ \forall u\in V, M>0$$ and now consider ...