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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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 ...
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1answer
77 views

Being compact is necessary for a continuous bijection to have a continuous inverse

Theorem: Suppose that $f:X \rightarrow Y$ is one-to-one, surjective and continuous. If $X$ is compact, Then $f^{-1}:Y \rightarrow X$ is also continuous. The proof for this theorem is pretty easy. ...
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1answer
71 views

Is a compact, bounded, closed subset of a normed vector space necessarily of finite dimension?

Let be X a Normed Vector Space, my question is: if a set A contained in X is compact, bounded, closed, is it finite dimension? I was looking for a characterization of the dimension of an nvs using ...
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1answer
85 views

Proof that this set $\{f\in X\mid \|f\|_\infty \le 1\}$ is not compact in $C[0,1]$ with the sup norm

Let $X=C[0,1]$ with the $\sup$ norm. Let $Y = \{f\in X\mid \|f\|_\infty \le 1\}$. It is my goal to show that $Y$ is not compact using the sequence defintion of compactness. Note that it is very easy ...
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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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245 views

Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$

Let $A \subset \ell^p$, where $1 \le p \lt \infty$. Suppose the following conditions are true: 1) $A$ is closed and bounded 2) $\forall \epsilon \gt 0, \: \exists \: N \in \mathbb{N}$ such ...
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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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31 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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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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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; $a$, $b$, ...
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57 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 $\nabla\mathbf{u}$ ...
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2answers
57 views

$E=\{e_k|k\in\mathbb{N}\}$ : characterization of $E$? [closed]

Let $(X, \|.\|)$ be normed linear space consisting of sequences $a=\{a(n)\}_{n=1}^{\infty}$ such that the series $\sum_{n=1}^\infty a(n)$ is absolutely convergent,with $ \|a\|= \sum_{n=1}^\infty |a(n)|...
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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
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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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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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 ...
2
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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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32 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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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
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1answer
32 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
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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 ...
4
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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 $\theta \in (0,1)$ ...
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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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1answer
40 views

Projections in the lp direct sum $E=(\bigoplus_{n=1}^\infty\ell_1^n)_p$.

Fix $1<p<\infty$ and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_1^n\right)_{\ell_p}.\end{equation*} Let $Y$ denote the closed subspace of $E$ ...
3
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1answer
51 views

Linear operator on a dense subset of $L_p$ which is unbounded when extended to $L_p$

Let the linear operator $L:C^{\infty}_0([-1,1]) \rightarrow C(\{0\})$ be defined as $f(0)$ (evaluating a function in $C^{\infty}_0([-1,1])$ at $0.$ I would like to show that extending this to ...
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1answer
37 views

Show that $||f||_{1,p}=||\dot{f}||_p$ is a norm on this space

For the space $C^{\infty}_0([-1,1])$ of continously differentiable functions on $[-1,1]$ with $f(-1)=f(1)=0$ I would like to show that $||f||1,p=||\dot{f}||_p$ is a norm on this space. I have ...
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1answer
105 views

A Continuous Function with a Divergent Fourier Series

This is a Q&A; I hope simply posting a question and then answering it is the right protocol. This is stuff I thought everybody knew, but in at least two recent threads it's turned out to be somewhat ...
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1answer
31 views

Normed and 2-normed spaces

Is every normed space a 2-normed space? Again, is every 2-normed space a normed space? please explain. Thanks in advance!
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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 ...
2
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1answer
33 views

Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
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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 ...
4
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2answers
179 views

explain why one can write $\hat{f}(\xi)=\lim_{n\to\infty}\frac{1}{\sqrt{2\pi}}\int_{-n}^{n}e^{-i\xi x}f(x)dx$ when $f\in L^2(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
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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= \...
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1answer
65 views

Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
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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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1answer
25 views

Tensor products $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$

Can anybody enlighten me, where the tensor products of the spaces of summable functions $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$ are ...
4
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28 views

Proof validation: complete set, change of variable

Let $\phi(x) \in \mathcal{C}^1([0,1])$ be a real valued function such that: $$\begin{cases} \phi'(x) > 0 & \forall x \in [0,1] \\ \phi(0) = 0 \\ \phi(1) = 1. \end{cases}$$ I'm asked to prove ...
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1answer
63 views

Topology for Hardy spaces

Let $\Omega\subset \mathbb{C}$ be an open set (of the complex plane) and let $\mathcal{H}(\Omega)$ be the algebra of analytic functions on $\Omega$ endowed with the topology of compact convergence (...
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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?
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2answers
765 views

Bounding $\int_0^1 f(x) dx $ under the condition $\int_0^1 f'(x)^2 dx \le 1$

Any tips on how to solve this? Problem 1.1.28 (Fa87) Let $S$ be the set of all real $C^1$ functions $f$ on $[0, 1]$ such that $f(0) = 0$ and $$\int_0^1 f'(x)^2 dx \le 1 \;. $$ Define ...
2
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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
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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: ...
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0answers
13 views

What is the difference between sequence space of bounded complex sequences compared to sequence spaces of bounded and unbounded complex sequences?

Both spaces have different measures as folllows (click to see) Bounded sequence space measure and Bounded and unbounded sequence space measure
2
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0answers
39 views

Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
0
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0answers
12 views

Extension of semi-norm in locally convex topological space

Suppose $X$ is a locally convex topological space,$M$ is a subspace. Suppose $p$ is a continuous semi-norm on $M$. Is it possible to extend $p$ to be a continuous semi-norm on $X$? What if $X$ is not ...
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2answers
38 views

Uniform closure of densely continuous functions

Consider the collection of those $\mathbb{R}$-valued functions on an interval $I\subseteq\mathbb{R}$, which have a dense set of points of continuity. I would expect this collection to be closed under ...
2
votes
1answer
37 views

Construct a sequence in Banach Space

Prove the equivalence between: $\forall x \in B_E = \{y \in E:\|y\| \leq 1 \}$ $\exists (x_n) \subset E$ such that $\|x_n\|=1$ and $x_n \rightarrow_w x$ (weak convergence). $\exists (x_n) \subset E$ ...
2
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1answer
26 views

SOT-isomorphic C*-algebras

Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras. Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that: $p_n \rightarrow 1$ in strong operator ...