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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Weak convergence implies boundedness in norm vector space

Let $(x_n)_{n\geq 1}$ be a sequence in the normed vector space $(X,\Vert\cdot\Vert)$ and let $x\in X$. Show that the following are equivalent. $x_n$ converges weakly to $x$ The sequence $(\Vert x_n \...
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2answers
35 views

Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: \begin{equation} h = \sum_n \langle h,h_n\...
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32 views

A Specific Problem on $\ell^1$ norm [on hold]

Let $$ B=\{\{x_n\}_{n=1}^\infty \in \ell^1 : \sum_{n=1}^\infty \frac{n}{n+1} x_n = 0\} $$ be a closed subset of $\ell^1$. Define $e = (1,0,0,\cdots)$. I want to prove that there is no $x\in B$ ...
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0answers
27 views

Proving Sobolev space on [0,1] is RKHS

My aim is to prove that the space: $\mathcal{H}$ = {$f:[0,1] \to \mathbb{R}: f\;is\;absolutely\;continuous,\;f(0)=f(1)=0,\;f'\in L^2[0,1]$} is a reproducing kernel hilbert space. Now assuming an ...
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0answers
36 views

Mathematical Definition of Entropy and a Question about the Nature of Stat. Mechanics Approach

I have been studying for quite some time now about entropic functionals, including Boltzmann-Gibbs, Renyi, Kaniadakis and Tsallis, and I am familiar with the properties that a functional has to ...
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1answer
13 views

Show that the Gelfand transform is a morphism?

Let $A$ be a commutative Banach algebra and let $x \in A$. We define the Gelfan transform of $x$ by $$\hat{x} (\chi)= \chi (x)$$ where $\chi$ is a nonzero multiplicative linear functional on $A$. I ...
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1answer
38 views

Prove that norm $\| y \|=\sqrt{\int_{a}^{b}| y(x)|^2 dx}$ forms a linear space

Prove that norm $$\mid\mid y \mid\mid=\sqrt{\int_{a}^{b}{| y(x)|}^2 dx}$$ forms a linear space. I am troubled at $$\mid\mid y+ \hat{y}\mid\mid\leq\mid\mid y\mid\mid + \mid\mid \hat{y}\mid\mid$$ ...
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1answer
14 views

Does the validity of Sobolev inequality on a domain imply it's a Sobolev extension domain?

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$ and let $1\leq p<n$. It is well--known that if $\Omega$ is an extension Domain for $W^{1,p}(\Omega)$, then the Sobolev inequality holds in that ...
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0answers
7 views

Concerning the Nonlinear Yosida Approximation

I am strugeling with this question for a while now and I hope that someone can help me. The setting is the following: Let $A:H\to\mathcal{P}(H)$ be a maximal monotone operator in the real Hilbert ...
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23 views

Weak compactness of closed convex subset

Let X be a reflexive Banach space. Is a closed convex subset of a weakly compact subset of X itself weakly compact?
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1answer
26 views

Why is $W^{k,p}(\Omega)$ the completion of $(\widetilde{C}^k(\Omega)$ instead of $W^{k,p}(\Omega) = (\widetilde{C}^k(\Omega)$?

In the first answer to this question the users states that if we define a norm $$\|f\|_{k,p} = \left( \sum_{|\alpha| \leq k} \|D^{\alpha} f\|_p^p \right)^{1/p},$$ and write $$\widetilde{C}^k(\Omega) = ...
4
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0answers
35 views

Integration of Hilbert space valued mappings.

TL;DR: Is there a version of the Bochner integral which allows for the integration of isometric embeddings $\phi:X\to H$ from a metric space to a Hilbert space, satisfying $\int_X \|\phi\| d\mu < \...
1
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1answer
17 views

Explanations for a “diagonal process” construction of a sequence

I am reading Peter Lax's "Functional analysis". Let $y_n$ be a bounded sequence of vectors in a Banach reflexive space, $X, Y$ their closed linear span. Take a countable set $m_j$ of applications ...
3
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1answer
62 views

A Question concerning $\ell^1$

Let $a_n >0$ and $\sum_{n=1}^\infty a_n < \infty$. Define $$ M=\{ \{x_n\}_{n=1}^\infty \in \ell^1 : \forall n, |x_n|\le a_n\} $$ Then I want to prove that $M$ is compact and $M$ cannot be ...
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1answer
38 views

Why are $C^0(\mathbb{R})$ and $C^{0,0} (\mathbb{R})$ the same spaces?

$C^0(\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})}$. $C^{0,0} (\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})} + \sup_{x,y \in \mathbb{R}, x \neq y} |f(x) - f(y)|$. I don't ...
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0answers
9 views

Is this correct for an explicit represenation of the $\Vert f \Vert_{C^2(\mathbb{R}^2)}$ norm that uses multi-index notation?

I'm not familiar with multi-index notation so I'm not sure if I have this correct. Say we have (taken from here) $$ \Vert f \Vert_{C^2(\mathbb{R}^2)} = \sum_{j=0}^2 \sup_{x \in \mathbb{R}^2} |\nabla^j ...
2
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1answer
38 views

Open sets and annihilator of functions

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
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2answers
55 views

$C([0,1])$ is separable: Is my solution correct?

Claim: $C([0,1])$ is separable w.r.t. the supremum norm. My solution: We want a countable subset $M$ s.t. $\forall f \in C([0,1])$ it exists a $g_n\in M $ s.t. $\lim_{n\to\infty}\Vert g_n-f \Vert_{\...
2
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0answers
19 views

continuous and sequentially continuous

If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous? And in the WOT/...
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0answers
16 views

Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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1answer
55 views

For $p(x)\in \Bbb{C}[x]$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $0\le k\le n-1$, show that $p(\lambda)=0\Rightarrow \lambda\in [0,1]$

For a complex polynomial $p(x)\in \Bbb{C}[x]$ of degree $n$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $1\le k\le n-1$, show that $p(\lambda)=0$ means $\lambda\in [0,1]$. I haven't come by any ...
2
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0answers
37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
2
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0answers
26 views

About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $ P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
0
votes
2answers
57 views

A necessary and sufficient condition for $B \subset \ell^1$ to be compact [duplicate]

I want to show that: For any $B\subset \ell^1$, $B$ is compact if and only if $B$ is bounded, closed and satisfies $$ \forall \epsilon>0, \; \exists N\in \mathbb{N}, \; \forall \{x_n\}_{n=1}...
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0answers
24 views

I want to show that the function space $C_0(X)$ is Banach [duplicate]

I'm reading some papers but I encountered a problem that "$C_0(X)$ is Banach space". Here $$ C_0(X):= \{ f: X\to \mathbb{C}: f \text{ is continuous and } \forall \epsilon>0, \exists K(\text{compact}...
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0answers
10 views

Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
0
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1answer
32 views

How to show this equality for operator norm?

Let $(X,\Sigma_\mu,d\mu)$ and $(Y,\Sigma_\nu,d\nu)$ be two positive $\sigma$-finite measure space and let $M(d\mu)$ and $M(d\nu)$ be spaces of complex-valued $d\mu$-measurable and $d\nu$-measurable ...
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1answer
24 views

Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
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1answer
23 views

How to think of Sobolev spaces $W^{k, p}$ for a function that is no longer an element of $W^{k, p}$ for $p$ greater than some number?

Consider the function $u(x) = x^{\frac{1}{2}}$ on the domain $[0, 1]$. This function is an element of $W^{1, 1}$ and $W^{1, \infty}$ but not $W^{1, 2}$ as for $W^{1, 1}$, we have $\Vert\frac{\...
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0answers
30 views

Closed Graph Theorem; Exercise

Let $E$ be a Banach space and let $T:E\to E^{\star}$ be a linear operator satisfying $<Tx,x>\geq 0$ $\forall x\in E$. Prove that $T$ is a bounded operator. My Solution (but I have trouble to ...
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votes
0answers
13 views

Different definitions of Besov norm/space

I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent. Victor I. Burenkov - Sobolev Spaces On Domains. Given $f:\mathbb{R}^n \...
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1answer
16 views

Weak and Weak$^{\star}$ topologies: Annihilator

Exercise: Let $E$ be a Banach space. Let $M\subset E$ be a linear subspace and let $f_0\in E^{\star}$. Prove that there exists some $g_0\in M^{\perp}$ s.t. \begin{equation}\inf_{g\in M^{\perp}}\Vert ...
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0answers
31 views

Functional analysis : show that the inf is attained

I'm a beginner in functional analysis and I'm trying to solve the following problem: $$ \alpha > 0,\;\; H^{\alpha}= \{ u \in \mathbb{R}^\mathbb{N} \ \mathrm{ such }\ \mathrm{that} \ \sum_{n=0}^{...
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8answers
2k views

Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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2answers
39 views

Every bounded linear operator $T$ between real Hilbert spaces is $T(x) = \sum \langle x,f_j\rangle\, e_j$

Let $T:H_1 \rightarrow H_2$, where $H_1$ and $H_2$ are real hilbert spaces and $T$ is a bounded linear operator. Prove the following: suppose $\{e_j\}$ an orthonormal basis for $H_2$, show that ...
2
votes
1answer
31 views

Convex basis of functions

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$? Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex ...
2
votes
1answer
17 views

Is $X^*$ complete with weak*-topology

Suppose $X$ is a topological vector space, $X^*$ is its topological dual space. Let the topology of $X^*$ is weak*-topology, Is $X^*$ complete? Suppose $f_s$ is a Cauchy net in $X^*$, it is easy to ...
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0answers
55 views

Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
4
votes
1answer
57 views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
6
votes
3answers
69 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
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1answer
48 views

Show all sequence of $l^1$ with $|x_n|\leq \frac{1}{n^2}$ is compact.

Could you help me to check my proof: let $\{x^k\}$ be a sequence in such set, we use Cantor's diagonal argument to show the existence of convergent subsequence. There exists a subsequence $\{x^{\...
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0answers
10 views

Questions about the regularity of the solution of the heat equation in a bounded domain

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
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3answers
81 views

$\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
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1answer
32 views

If $\mathcal H$ is the closure of the set $D$ of divergence-free smooth functions in $L^2$, then $H_0^1∩\mathcal H$ is the closure of $D$ in $H_0^1$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$\mathfrak D:=\left\{u\in\mathcal D:\nabla\cdot u=0\right\}$$ $H:=H_0^1(\Omega,\mathbb R^d)$...
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vote
1answer
37 views

What is the Hilbert adjoint operator of this bounded linear operator?

Let $H$ be a Hilbert space, and let $z \in H$. Let $T_z \colon H \to K$, where $K$ is the field of scalars for $H$ and $K$ is either $\mathbb{R}$ or $\mathbb{C}$, be defined by $$ T_z (x) \colon= \...
0
votes
0answers
54 views

Two inner products in one vector space.

Please, can you help me answer the some following questions? In theory functional analysis. At first, I want to consider finitely dimensional vector space V over field K(real or complex). Now, if it ...
2
votes
0answers
16 views

Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
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votes
1answer
36 views

A basis of $L^2$

I would like to ask you a question that is there a basis of the space $L^2(\Omega,\mathcal{F},\mathbb{P})$, where $\mathbb{P}$-probability measure?
2
votes
0answers
44 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
2
votes
1answer
22 views

Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...