Tagged Questions

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, ...

learn more… | top users | synonyms (1)

0
votes
0answers
20 views

Convergence uniformly.

Let $\Omega$ a domain bounded and supoose that $u^{\epsilon}(x)\rightarrow u(x)$ uniformly. How do I show that $$ \int_{\Omega}|\nabla u^{\epsilon}|^2\rightarrow\int_{\Omega}|\nabla u|^2? $$
1
vote
2answers
27 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
4
votes
1answer
39 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
3
votes
0answers
9 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
3
votes
1answer
19 views

Positivelinear operator on $L^p$-spaces

Suppose $1<p<\infty$. A linear operator $T \colon L^p(\Omega)\to L^p(\Omega)$ is positive if $f \geq 0$ imply $T(f)\geq 0$ (where $\Omega$ is a measure space). 1) Does there exist a positive ...
0
votes
1answer
22 views

The topology on $C^\infty_c(\mathbb{R}^d)$ used for “distributions of compact support”

On the one hand, Eskin's book on PDEs tells me that I should be content to think of this topology as one "described" (not fully, and it's not even clear it's a topology) by the convergence of ...
0
votes
1answer
22 views

Convergence of a sequence of linearly independent vectors in normed space

In an infinite dimensional normed vector space is it possible to find a sequence ${v_n}$ of linearly independent vector (so the sequence is a set of linearly independent vectors) each has norm 1 such ...
0
votes
0answers
11 views

A set of differential forms, uniformly bounded with their Laplacians, is precompact in $L^2$.

Let $M$ be a compact Riemannian manifold and let $\Delta$ be a Hodge Laplacian on $k$-forms. How to show that the if the set $\{u_\alpha\} \subset C^2(M,\Lambda^k)$ of $C^2$ $k$-forms is uniformly ...
3
votes
2answers
41 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
1
vote
1answer
30 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
2
votes
1answer
25 views

Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
1
vote
1answer
6 views

product of spaces of bounded linear operators

Let $E$ be a normed space. Let $(F_i)_{i \in I}$ be a family of normed spaces. Show that $\prod_{i \in I}{\mathcal{L}(E, F_i)}$ and $\mathcal{L}(E,\prod_{i \in I}F_i)$ are isometrically isomorphic. ...
0
votes
0answers
18 views

Sobolev norm inequality.

I would like to prove or to disprove the following statement. Let $u$ and $v$ be functions in $H^{s}(S^1)$, the for every $s'\leq s$ $$\|uv\|_s\leq (\|u\|_{s}\|v\|_{s'}+\|v\|_{s}\|u\|_{s'}).$$ I ...
0
votes
1answer
13 views

annihilator subspace of normed space

Let $E_0$ be a subpace of the normed space $E$. Let $E_0^a = \{f \in E' : f(x)=0 \forall x \in E_0 \}$ $(E'=\mathcal{L}(E,\mathbb{K})$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C})$. Show that ...
1
vote
2answers
25 views

Estimate in Sobolev Spaces

Let $u\in H^0(U)\cap H^1_0(U)$ and $v_k\in C^\infty_c$(U) such that $v_k\rightarrow u$ in $H^1_0(U)$ and $w_k\in C^\infty (U)$ such that $w_k \rightarrow u $ in $H^2(U)$. I want to show that $ \int_U ...
0
votes
2answers
29 views

Dual space of a finite dimensional normed space

My lecturer gave us this result today in class, but he didn't give a proof, he said we can prove it ourselves, only I'm really struggling to see how to do it. Let $E$ be a normed space with dual ...
1
vote
1answer
12 views

How to create distribution function from sketch?

I'm playing with image manipulation based on various mathematical algorighms (such as edge detection). I'm also changing the colors in various ways just to see what comes out of it. Regarding this, ...
1
vote
1answer
29 views

Practical convergence in $C^{\infty}_c$

Let $C^{\infty}_c$ be the space of $C^{\infty}$ functions with compact support in $\mathbb{R}$ with the usual topology derived by the convergence in infinity norm in every $C^{k}_c$. I would like to ...
0
votes
0answers
12 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finit, with ...
3
votes
1answer
35 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
4
votes
1answer
39 views

A question about sublinear functionals

Could you please give me hints may leads to prove the following: Let $X$ be a real vector space, $\,p_1,p_2:X\to\mathbb R\,$ be two sublinear functionals, and $\,f:X\to\mathbb R\,$ be a linear ...
2
votes
1answer
25 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
4
votes
1answer
26 views

Extension of the limit operator on $l^\infty$

Let $l^\infty = \{x\in \mathbb{R}^\mathbb{N}\colon \sup_{n\in \mathbb{N}}|x_n|<\infty\}$ and the subspace $C \subseteq l^\infty$ given by the convergent sequences. We consider the linear operator ...
0
votes
0answers
10 views

Equivalence of lens shaped domain and the existence of a smooth time function

A lens shape doimain is defined here as: Defn A lens-shaped domain $D\subset M$ based on $\Sigma$ is the image of a smooth map $\Phi: \Sigma\times (-1,1) \to M$ where $\Sigma\subset M$ is a compact, ...
0
votes
1answer
21 views

Is this a valid operator norm?

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm? (I think it is. As it satisfies ...
0
votes
0answers
25 views

Definition of reflexive Banach spaces

I'm trying to understand the definition of reflexive spaces. I wrote in my notes: If $Y$ is reflexive then for all $\eta\in Y^{**}$, $f\in Y^*$, $\exists y\in Y$ where $\eta(f) = f(y)$. My question ...
0
votes
0answers
30 views

If $T_n:H\to H$ ($n=1,2,\dots$) are normal linear operators and $T_n\to T$, show that $T$ is a normal linear operator. [on hold]

If $T_n:H\to H$ ($n=1,2,\dots$) are normal linear operators and $T_n\to T$, show that $T$ is a normal linear operator. This seems obvious, so I don't know how to go about showing it. If $T_n:H\to H$ ...
1
vote
2answers
52 views

About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
1
vote
2answers
11 views

Show that the matrix $(a_{j,k})_{j,k\in \mathbb{N}}$ induces a bounded operator on $\ell^2$.

I have a matrix $(a_{j,k})_{j,k\in\mathbb{N}}$ given by: $ a_{j,k} = \dfrac{1 -e^{-jk}}{jk + 1}$ and I need to show that this induces a bounded operator on $\ell^2$. I'm pretty sure Schur's test is ...
0
votes
1answer
13 views

what is permutation invariant sequence space

what is permutation invariant sequence spaces? And why $c_{00}$ is the smallest permutation invariant sequence space?
1
vote
1answer
20 views

canonical form of finite rank operators

Let $X,Y$ be banach spaces and let $T:X\rightarrow Y$ be an linear continuous operator with finite dimensional image $Im(T)\subset Y$. Now I want to prove that there exists continuous linear ...
7
votes
0answers
40 views

Mankiewicz theorem

I'm looking for a proof of Mankiewicz theorem, which states that: If $U, V$ are open, connected subsets of normed spaces $E, F$ respectively, then every bijective isometry $U \rightarrow V$ extends ...
2
votes
3answers
35 views

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$.

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$. I've been stuck on this for a while and don't really know where to start.
0
votes
1answer
12 views

If $S$ and $T$ and bounded self-adjoint linear operators on a Hilbert space $H$, show that $\tilde{T}=\alpha S +\beta T$ is self-adjoint.

If $S$ and $T$ and bounded self-adjoint linear operators on a Hilbert space $H$ and $\alpha$ and $\beta$ are real, show that $\tilde{T}=\alpha S +\beta T$ is self-adjoint. Can't figure out where to ...
1
vote
1answer
36 views

Showing an operator is essentially self-adjoint

I have a question about checking if an operator is essentially self-adjoint. Given the operator $$H=-\frac{1}{2}\partial^2_{r}-\frac{1}{r}\partial_r$$ with domain $C^{\infty}_0((0,\infty))$ (i.e. ...
0
votes
0answers
28 views

Is the sequence convergent? [on hold]

Please give me a detailed explanation of this, consider me a layman in functional analysis! This question requires me to find whether the sequences are convergent in nature, using the following data. ...
1
vote
1answer
14 views

$\mathcal{V}$ attains a minimum

Let $V: \mathbb{R}\rightarrow [0, \infty)$ be convex, continuous and s.t. $\exists\, p \in (1,\infty), \delta >0$ with $V(t) \geq \delta \lvert t \rvert^p, \; \forall t \in \mathbb{R}\setminus ...
3
votes
0answers
56 views

Is exponential function in a C*-algebra injective on self-adjoint elements?

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?
0
votes
1answer
48 views

an example of a sequence $(u_n)_n$ taking its values in $[-1,+1]$ such that $(u_{n+1}-u_n)$ converge to zero but $(u_n)_n$ does not converge

Define a sequence $(u_n)_n$ by: $$u_n=\cos(\log n).$$ Then, it is easy to show that $(u_{n+1}-u_n)$ goes to zero at infinity. The question is how to prove that $(u_n)_n$ is a divergent sequence ...
1
vote
0answers
22 views

Relative Entropy and variation formula for $C_c$

Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$. By the varational representation formula of Donsker and Varadhan we know ...
0
votes
0answers
20 views

How to define probability density function in Hilbert space??

Consider the space of random continuous functions $f:(0,1)\rightarrow\mathbb R$. Suppose we assume that this is a Hilbert space. Is there any notion of probability density function in the Hilbert ...
1
vote
0answers
18 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...
10
votes
2answers
269 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
5
votes
2answers
84 views

Inverse of an infinitely large matrix?

This is probably a trivial problem for some people, but I've spent quite some time on it: What is the inverse of the infinite matrix $$ \left[\begin{matrix} 0^0 & 0^1 & 0^2 & 0^3 & ...
0
votes
0answers
8 views

Dieudonné separation theorem for locally $L^{\infty} $-convex modules

Does anybody knows if there exists a version of the Dieudonné separation theorem for locally $L^{\infty} $-convex modules? I know that such a theorem exists for locally $L^{0} $-convex modules, but ...
2
votes
1answer
32 views

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
0
votes
1answer
19 views

Adjoint operator between Hilbert spaces, does it map a subspace onto a subspace?

Let $X$ and $Y$ be Hilbert spaces and let $f\colon X \to Y$ be a linear continuous bijection. Define the adjoint $f'\colon Y \to X$ by $(f'y, x)_X = (y, fx)_Y$. Let $X_0$ and $Y_0$ be subspaces of ...
0
votes
0answers
35 views

Weak convergence and convergence in distribution

Is convergence in distribution related to weak convergence in Banach theory? Where by weak convergence I mean: for every functional f the sequence $\langle f,x_n\rangle \overset{n}{\rightarrow} ...
4
votes
1answer
53 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to\infty.$$ Obviously, this sequence of functions ...
1
vote
1answer
48 views

Measurable functional calculus

I am struggeling with this exercise: Let $T \in L(H)$ be a self-adjoint operator and $\Psi$ be a measurable (Borel) functional calculus on the spectrum of $T$. For a Borel set $\Delta \subset \sigma ...