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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Why does a real spectrum not have holes?

Apparently, if $A$ is a $C^\ast$-algebra and the spectrum $\sigma(a) \subseteq \mathbb R$ then $\sigma (a)$ does not have holes. I read this in Murphy and then tried to prove it since I didn't ...
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23 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
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70 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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50 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
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1answer
29 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
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1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
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1answer
25 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
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61 views

$C([0,1])$ is not weakly sequentially complete.

I'm studying Functional Analysis by myself. For a counterexample of every Banach space is not weakly sequentially complete, I was suggested to check $C([0,1])$ is not weakly sequentially complete. For ...
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0answers
29 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: Let $E$ be a measurable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. Suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
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2answers
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$L^1(μ)$ is finite dimensional if it is reflexive

If $(X,\Omega,\mu)$ is a $\sigma -$ finite measure space, show that if $L^1(X,\Omega,\mu)$ is reflexive then it is finite dimensional. My attempt: I want to show there is a copy of $\ell^1$ in ...
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1answer
31 views

How to get this inequality given another inequality (norms and logarithms)

Suppose I have proved for all $f$ positive with $\lVert f \rVert_{L^2} = 1$ that $$\int f^2\log(f) \leq -C_1\log(\epsilon) + \epsilon\lVert \nabla f \rVert_{L^2}^2.$$ I want to prove that for all $f$ ...
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1answer
38 views

Does $\|a\|^2 \le \|a^\ast a\|$ imply $\|a^\ast \| \|a\| \le \|a^2\|$?

Let $A$ be a normed algebra with an involution $\ast$. As per title, my question is whether $\|a\| \|a\| \le \|a^\ast a\|$ implies $\|a^\ast \| \|a\| \le \|a^2\|$. I used it in a proof I wrote ...
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0answers
35 views

proof of the existence of spherical sections of ellipsoids

i want to prove : Let L be proper ellipsoid with the origin as center in $E^{2m-1}$ .There exists a subspace $E^m$ such that $E^m$ intersects $L$ is an m-dimensional sphere it is proven by Dvoretzky ...
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1answer
24 views

A question on nuclearity

Definition 2.1.1. If $A$, $B$ are C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
2
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1answer
51 views

A simple lemma in tensor product

Here is a quotation of a book: ($\otimes$ denotes the minimal tensor product) Lemma 3.9.2. Let $A$ be a C*-algebra. If $E\subset A$ is an operator system and $J\triangleleft B$ is an ideal, then ...
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0answers
20 views

Proving that integrator operator of a kernel satisfies a specific peroperty

I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such ...
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1answer
27 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
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1answer
52 views

What does it mean to ''construct a Riemann-Stieltjes integral''?

This question refers to an exercise (1.11c) from Reed & Simon's book on mathematical physics. In the first parts of this problem, the reader is introduced to the notion of functions of bounded ...
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50 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
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0answers
13 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
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0answers
23 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
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1answer
102 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
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1answer
26 views

Why is it a borel set on the boundary of the unit ball of $E^n$?

Given $C$ convex body (compact convex set with non-empty interior points) in $E^n$ symmetric about the origin and containing the unit ball. Let $A(r)$ denote ,for every real $r >1$, the subset of ...
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1answer
32 views

Linear operator determined by its action on a complete orthonormal sequence

Let $(e_n)$ be a complete orthonormal sequence in the Hilbert space H and let $(\lambda_n)$ be a sequence of scalars, not necessarily bounded. I'm trying to prove that the operator defined as $$Tv = ...
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1answer
43 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
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0answers
40 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
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1answer
36 views

what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...
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2answers
52 views

Why locally compact hausdorff space

I may be missing something very trivial,but cannot figure it out. To prove the fact that the space $C_0(X)$ is a Banach space under the usual sup norm when $X$ is locally compact and Hausdorff, where ...
2
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1answer
47 views

How is equivalent to approximation property?

I'm trying to prove Lemma 4.8 of [1] online reading: Notation: $\tau_C(X)$ means the topology of uniform convergence on the compact subsets of $X$. Lemma 4.8. For a Banach space $X$ the following ...
3
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1answer
32 views

Closed subspace of a reflexive Banach space is reflexive

I'm studying Conway's functional Analysis by myself. In page 132 of his book, for showing every Closed subspace M of a reflexive Banach space X is reflexive, he says ...
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0answers
24 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
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2answers
43 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
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1answer
32 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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1answer
13 views

For bounded operator $U$, show that if $UU^*$, $U^*U$ are projections, then $U$ is a partial isometry

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $U : \mathcal{H} \to\mathcal{H}$ is a bounded linear operator such that $UU^*$ and ...
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0answers
37 views

Sequences in $L_2$ space

Let $M$ be compact metric space, with finite full measure $\mu.$ Let $\epsilon:M\to\mathbb{R}$ be a continuous function. Assume that $\epsilon>0$ in $M\setminus \{p_0\}$ for some $p_0\in M$ and ...
3
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1answer
42 views

E is bounded if and only if every countable subset of it is bounded

I'm trying to do an exercise from Rudin's "Functional analysis, 2nd edition". It is question 6 from the first chapter: "Prove that a set E in a topological vector space is bounded if and only if ...
4
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2answers
44 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
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0answers
118 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
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1answer
22 views

Prove that the Besov Space is a Banach space

Help me prove that the Besov space is a Banach space. I need to show that the Besov space is complete. If the Besov space is a closed subset of $L_p$ and since all $L_p$ spaces are complete then I'm ...
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1answer
31 views

Question about different defintions of isometry on a Hilbert space

Let $(\mathcal{H} , (\cdot, \cdot))$ be a Hilbert space over the field $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$ (so the norm on $\mathcal{H}$ is given by $\|\cdot\| = (\cdot, \cdot)^{\frac{1}{2}}$). ...
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0answers
37 views

Taking a limit but everything is only almost everywhere

Suppose $h$ is a function such that $h(s)$ exists for all $s$ and $\lim_{s \to t} h(s)$ exists for all $t$. Here $s$ and $t$ range over some bounded interval. Suppose for a.e. $t$ and a.e. $s$, we ...
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1answer
25 views

Closure of a normed space

Every normed space $A$ is isometrically isomorphic to a dense subset of a Banach space $B$. Is it okay for us to view $A$ as a subspace of $B$ and hence say that "the closure of $A$ is Banach", or am ...
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1answer
21 views

Bijective transformation of $L^2$-weak convergence sequence again weak converging?

Let $f_n$ converge weakly in $L^2(x)([0,1])$ to $f$, with $|f_n(x)|\leq C$ for almost all $x\in]0,1]$ and all $n$. Let $H:R\rightarrow R$ be strong monotone increasing and continuous with $H(0)=0$. ...
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1answer
39 views

Does weak convergence in $L^2$ implies convergence almost everywhere along subsequence?

If I know $\int_{[0,1]} f_{n}(x) g(x) dx \rightarrow \int_{[0,1]} f(x) g(x) dx$ as $n \rightarrow \infty$ for all $g \in L^2([0,1])$ (weak convergence in $L^2$) and $|f_n(x)|_{L^2} <C$ (uniformly ...
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0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
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1answer
23 views

dual space of a locally convex space

I'm self studying Rudin's Functional Analysis by myself. In page 68, he wants to show that $(X^*,w^*)^*=X$ where X is a topological vector space and whose dual is $X^*$. He says every $x\in X$ ...
3
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1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
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0answers
43 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
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0answers
29 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
0
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2answers
72 views

Looking for a book: $B(H)$ not reflexive

I'm looking for a book with a proof that for an infinite dimensional Hilbert space, $B(H)$ is not reflexive. Thank you.