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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17 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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29 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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1answer
25 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
16 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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1answer
23 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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15 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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2answers
44 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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0answers
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
13 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
13 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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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 ...
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42 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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0answers
25 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
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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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 ...
2
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1answer
29 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 ...
3
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1answer
32 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 ...
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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 ...
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0answers
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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1answer
34 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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0answers
33 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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31 views

Existence theorems depending on compactness of unit ball? [on hold]

I can only think of that a semi-continous function attain it's maximum on compact sets. What other existance themorems depend on compactness of unit ball? Which cases are we able to maintain and which ...
8
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1answer
125 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
67 views

An equality in Hilbert spaces

To understand a proof in functional analysis I need to understand why the following equation is true: $$\lVert x\rVert^2 - \sum_{j=1}^n |x_i|^2 = \Biggl\lVert x-\sum_{i=1}^nx_ie_i\Biggr\rVert^2$$ ...
3
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1answer
33 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 ...
2
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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 ...
2
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1answer
34 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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2answers
63 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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9 views

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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0answers
16 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
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: ...
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1answer
37 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 ...
2
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1answer
50 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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43 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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0answers
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 ...
2
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1answer
34 views

Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
2
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2answers
28 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
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1answer
36 views

Continuous linear operator T at a point T is then continued

The problem is the next If T is continuous at a single point, it is continuous, without using that T is continuous iff T is bounded. I tried this result as follows If T is continuous at a single ...
3
votes
1answer
37 views

Strict positiveness on a C*-algebra given by generators and relations.

Let $A$ be a C*-algebra with generators $a_1,a_2,\ldots,a_n$ and some (non-important) relations (the relations imply that $\|a_i\|\leq 1$, so that $A$ exists). Among the given relations we have that ...
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42 views

A convergence problem for a sequence of functions

Let $\{f_n\}$ be a sequence of functions in $L^p$ with $0<p<1$ let $b>1$ and $(b^n(f_{n+1}-f_{n})\rightarrow 0)$ what we can say about $\{f_n\}$ ?is it Cauchy?
3
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1answer
35 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
1
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1answer
21 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
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0answers
23 views

Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [on hold]

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
2
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1answer
38 views

Sobolev space on a closed subset

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a bounded open subset and $dx$ be a Lebesgue measure on $U$ \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
0
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1answer
19 views

Infinite sum of bounded linear operators on a Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, separable, complex Hilbert space, and let $\mathbf{a}$ and $\mathbf{b}$ be bounded linear operators on $\mathcal{H}$ such that ...
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votes
2answers
27 views

Locally boundedness of some L^p spaces.

It is well known that $L^p$ spaces for $ 0<p<1 $ are not locally convex. I would like to know whether they are locally bounded.
0
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1answer
42 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
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1answer
25 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...