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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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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24 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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12 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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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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2answers
105 views
+200

Radon-Nikodým (write the density as a limit)

Let $\mu$ be a probability measure and $\nu$ a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B})$ with $\nu\ll\mu$. Show that it is $\mu$-a.s. $$ \lim_{h\to 0}\frac{\nu [x-h,x+h)}{\mu ...
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1answer
39 views

how shall i'll prove if c 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.
6
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1answer
41 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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1answer
50 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 ...
2
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2answers
56 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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28 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than $\mathfrak c$ can be complete. Can we prove it without using AC or the Hahn-Banach Theorem?
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28 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 ...
1
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0answers
27 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 ...
0
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1answer
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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0answers
39 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 ...
3
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0answers
22 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: ...
3
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1answer
83 views

Soft question: What are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
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4 views

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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0answers
14 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 $F(L)$ -defined by the functional calculus ...
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0answers
14 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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0answers
23 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 ...
4
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1answer
341 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
0
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3answers
55 views

Selfadjoint Operators: Sesquilinear Form (II)

Given a Hilbert space $\mathcal{H}$. Consider a positive form: $$s:\mathcal{D}\to\mathcal{H}:\quad s(\varphi,\varphi)\geq0$$ Introduce its form space: ...
3
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1answer
2k views

Inner product is jointly continuous

I'm attempting another exercise from my notes: Show that an inner product on an inner product space is jointly continuous with respect to the induced norm:if $v_n \to v$ and $w_n \to w$ as $n \to ...
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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 ...
2
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2answers
31 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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1answer
38 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
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1answer
45 views

How do you show that $||.||: N \to \mathbb{R}$ defined as below defines a norm on $N?$

We are give $||x|| \ge 0$ for all $x \in N$ and $=0$ iff $x=0$, and $||\alpha x||= |\alpha | ||x||$ for all $x \in N$ and scalars. So we have that the first and seconds condition for a norm holds. I ...
2
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1answer
30 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 ...
0
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1answer
42 views

Normal Operators: Transform (III)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$N=Z\sqrt{1-Z^*Z}^{-1}$$ Especially one had: ...
3
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1answer
33 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 ...
2
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1answer
58 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 ...
2
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1answer
53 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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0answers
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 ...
6
votes
2answers
344 views

For a multiplication operator $M_f$ on $L^2$ with $f\geq 0$, is $SM_fS^{*}$ positive?

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
4
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1answer
673 views

For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
1
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1answer
135 views

Does inversion reverse order for positive elements in a unital C* algebra?

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related ...
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1answer
15 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: ...
1
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1answer
38 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 ...
8
votes
1answer
128 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 ...
3
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3answers
68 views

An equality in Hilbert spaces

To understand a proof in functional analysis I need to understand why the following equation is true: $$\lVert x\rVert^2 - \sum_{j=1}^n |x_i|^2 = \Biggl\lVert x-\sum_{i=1}^nx_ie_i\Biggr\rVert^2$$ ...
5
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1answer
39 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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2answers
32 views

Is range a of a generator of a strongly continuous semi group in doman of the generator?

Let $X$ be a banach space and $A:D(A)\rightarrow X$ be a generator of a infinte seminal generator of a $C_0$ semi group $\{S(t)\}_{t\geq 0}$. In this case is it possible that ...
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1answer
34 views

Predual: Denseness

Problem Given a Banach space $E$. Regard a subspace: $$\iota:U\hookrightarrow E:u\mapsto u$$ Consider the projection: $$\pi:E'\twoheadrightarrow U':\psi\mapsto\psi\circ\iota$$ By Hahn-Banach find: ...
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1answer
37 views

Describe the GNS construction [closed]

Question: Describe the GNS construction for the C$^*$-algebra $ C[0, 1]$ and for the positive linear functional $\phi $ given by $\phi(f) = f (0)$. What should i do? Should I describe Hilbert space ...
0
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1answer
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 ...
3
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2answers
65 views

There are $u$ in $W^{1,p}(D)$ and a weakly converging subsequence $\left\{ u_{m_{k}}\right\} $ to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
3
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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 ...
1
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1answer
42 views

Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{K}:\quad A=A^{**}$$ Construct its modulus: ...
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1answer
73 views

Spectral Measures: Polar Decomposition

Isometric Equality Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$ It gives rise to operators: ...
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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 ...