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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Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
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552 views

A subset of a finite dimensional normed vector space

This was left as an unproved theorem in our class: Theroem: If $X$ is a finite dimensional normed vector space then each subset $M$ of $X$ is compact if and only if $M$ is closed and bounded. How ...
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659 views

Closed unit ball in C0(R) has no extreme points

I am trying to show that the closed unit ball in $C_0(R)$ has no extreme points. This is what I got so far and I am stuck. Please help me. Suppose that $f \in C_0(R)$ and is an extreme point of the ...
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195 views

Exercise:Bounded linear operator

Assume $\alpha(.)$ is a function defined on the bounded measurable set $E$.Let $$(Tx)(t)=\alpha(t)x(t)\ \ \ \ \ \ \ \ \ \ \ \ x\in L^2(E)$$.Then $T$ is a bounded linear operator from $L^2(E)$ to ...
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713 views

Continuous linear mapping and bounded subsets

Continuous linear mappings between topological vector spaces preserve boundedness. I was wondering if it means that the inverse image of a bounded subset under a continuous linear mapping is ...
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538 views

Angle between functions

I have a rather simple question but googling it did not bring a satisfactory result: Assume you have given two function $f$ and $g$ on some space $\mathcal{L}^2(\Omega)$ where $\Omega \subset ...
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622 views

Is this functional convex?

Maybe this is a strange or un-professional question. I want to know whether my equation is convex. My equation is as follows: ...
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264 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...
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408 views

Strict convexity of a norm on $C[0,1]$

Why the following norm of $C[0,1]$ is strictly convex $||| f|||= ||f||_\infty + ||f||_{L^2 [0,1]}$ where $||f||_{L^2 [0,1]}$ refers to $p$-norm of $L^p[0,1]$ when $p=2$. I'm not sure if this ...
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1k views

Why is the $L_p$ norm strictly convex for $1<p<\infty?$

Let $x,y \in L_p$ such that $\|x\|_p=\|y\|_p=1$ , $1< p<\infty$ and $x\neq y.$ Why is $\|x+y\|_p<2$ ? I'm not sure how to start the proof.. I don't know how to handle integral of ...
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452 views

Complete orthonormal sequence, Hilbert Space, Kronecker Delta

Let $H$ be a Hilbert space and $(e_n)_{n=1,2,\ldots}$ be a complete orthonormal sequence in $H$. We want to show that if $a_{np}=(e_n,f_p)$ then $\sum_{p=1}^{\infty}a_{np} ...
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1answer
141 views

Extend an isomorphism

Let $E$ and $E'$ metric spaces. If $E$ isomorphic to a subspace of $E'$, then $E'$ is isomorphic to an space that contain $E$.
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3answers
167 views

How can I write $\frac{1}{(a+x)}$ as an exponential function $y = Ce^{-kx}$?

How can I write $\frac{1}{a+x}$, $a$ a non-zero positive constant, in exponential terms in the form of $y = Ce^{-kx}$? I've tried to use to Taylor series but they only seem to work for $x < 1$.
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341 views

Surjectivity implying injectivity for bounded linear operators

Suppose $X$ is a Banach space and $T: X \to X$ is a bounded linear operator. If $T$ is onto, is it necessarily 1-1?
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575 views

Relation between the norms on X and Y and the induced product norm on X x Y

Let $(X,||\cdot||_X)$, $(Y,||\cdot||_Y)$ be a pair of normed linear spaces, and $(X \times Y, ||\cdot||_{X \times Y})$ the induced product space and norm. If $(x,y)$ is an element in $X \times Y$, is ...
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27 views

Problem with the Definition of contractible set

I have this definition of contractible set: we say that $A\subset X$ is contractible in $X$ if there exists a continuous function $\eta:[0,1]\times A\rightarrow X$ such that $\eta(0,x)=x, \forall ...
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60 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
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34 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
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29 views

Show that properties of norm are satisfied

Show that \begin{align} & \|y\|_M= \max_{a \leq x \leq b} |y(x)| \tag 1 \\[8pt] & \|y\|_1=\int_a^b |y(x)|\, dx \tag 2 \end{align} satify the properties of a norm in $C[a,b]$. That's what I ...
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35 views

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$. My intuition is to use Young's Inequality and then apply it to $A_k=\frac{|x_k|}{\|x\|}$ ...
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1answer
13 views

Hermitian operator and numerical range

How to prove that for a complex Hilbert space $\mathcal H$ an operator $T:\mathcal H \to \mathcal H$ is hermitian if and only if it's numerical range $W(T)$ is real, where $W(T)=\{\langle Tx,x \rangle ...
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17 views

Topology induced by seminorms and initial topology

Let's say we have a family of seminorms $(\rho_\alpha)_{\alpha \in A}$ on a vector space $V$. There are two ways to topologize $V$ using those seminorms: We define topology $\mathcal S$ by a ...
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47 views

Space of functions on $[a, b]$ with countable discontinuity

We are quite familiar with $C([a,b])$, the set of all continuous functions defined on a closed interval $[a,b]$ with the supremum norm. My question is; if we allow function with countable ...
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1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
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28 views

Functional analysis-

Consider the Banach algebra $A=C_{b}(K)$ of all complex-valued bounded continuous functions on a completely regular Hausdorff space $K$ with the supremum norm, and let $C$ be the set $C:=\{g \in A: ...
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65 views

How do I prove a differential operator has no purely imaginary eigenvalues?

Anyone who has taken a course in linear algebra knows how to prove the eigenvalues of a self-adjoint operator are real or the eigenvalues of a skew-self-adjoint operator are purely imaginary. This is ...
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1answer
19 views

ONB of Hilbert dual $H'$

Let $H$ an arbitrary Hilbert space, $\{ e_i \}_{i \in I}$ ONB of $H$. Is there an ONB $\{ e^j \}_{j \in I}$ of the Hilbert dual $H'$, s.t. $e^j(e_i)=\delta_{ij}$? If so, is $\{e_i \otimes e^j\}_{i,j ...
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1answer
36 views

Evaluate the spectrum of a bounded linear operator

$H$ is a separable Hilbert space over $\mathbb C$ and $\{u_n\}$ is a maximal orthonormal set of H. $A \in B(H)$ and there exists $\lambda \in \mathbb C$ such that $$A(u_n) = \lambda u_n - u_{n+1}, n = ...
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27 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
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1answer
95 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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1answer
43 views

Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
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1answer
50 views

Stone Weierstrass and Runge

Suppose $E(closed)\subset\{z:|z|=1\}$ and let $f(z)$ be a continuous function on the set $E$. I want to show that $f(z)$ can be approximated by polynomials on $E$. I am not exactly sure how to solve ...
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1answer
50 views

Is the Inner Product a uniformly continuous function?

I know it's continuous but is it uniformly continuous?
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75 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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47 views

Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.

I've been stuck on this question for a long time now and was wondering if anyone could show me how it's done. So far I have done the following: Since $\lim_{x\to \infty}f'(x)=M$ then $\forall \epsilon ...
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2answers
45 views

Misunderstanding a result from functional analysis

While reading page 111 of this book I got confused as to what the authors were doing in their counterexample of why strong convergence doesn't imply uniform convergence. I summarise it below Let ...
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1answer
33 views

What's the spectrum of the operator $g\longmapsto f\cdot g$?

What is the spectrum of the operator: $$T: C[0, 1]\longrightarrow C[0, 1], g\longmapsto f\cdot g$$ where $f\in C[0, 1]$ is a fixed function? Here I'm considering the space $C[0, 1]$ endowed with the ...
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1answer
33 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
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42 views

A dense subspace of L^2

Let $\mathcal{H}$ be the Hilbert space of holomorphic functions defined on the unit disc $D\subset\mathbb{C}$ which is the clousure of the complex polynomial functions on the disc with respect to the ...
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1answer
47 views

how to show the derivative of the polynomial is bounded by itself in certain space.

How to prove that for every positive integer $d$, there exists $C(d)>0$, such that: For every polynomial with degree $\leq d$, we have $\max\limits_{x\in [0,1]}|p'(x)|\leq C(d)\max\limits_{x\in ...
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1answer
26 views

three simple question about Banach space

If $X$ is a Banach space, and its closed unit ball is separable, then do we know that $X$ is separable? If $X$ is a separable Banach space, then do we know that its closed unit ball is separable? ...
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1answer
28 views

Show that $\left\Vert f\right\Vert _{p}\leq\left\Vert f\right\Vert _{\infty}$ for all $1\leq p<\infty$

My questión is simple: Let $f\in L^{\infty}\left(\mu\right)$ whit $\mu$ a probability. Show that $\left\Vert f\right\Vert _{p}\leq\left\Vert f\right\Vert _{\infty}$ for all $1\leq p<\infty$. ...
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47 views

Bounded linear map on topological vector spaces is continuous

Let $X$ and $Y$ be topological vector spaces and $T\colon X\to Y$ linear. Suppose that $T$ sends bounded sets to bounded sets and that $X$ is first countable. The claim is that $T$ is continuous. ...
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44 views

Pointwise convergence and absolute convergence in $L^1([0,1])$ imply uniform convergence?

Consider a pointwise convergent series $\sum f_n$ of continuous (hence also bounded) functions on $[0,1]$, which is absolutely convergent in $L^1$. Isn't the series also uniformly convergent in this ...
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1answer
27 views

How to calculate the adjoint of an operator and its domain?

Let $A : D(A) \subset L^2(0,1) \to L^2(0, 1)$, $$D(A) = \{u \in H^2([0, 1]) : u(0) = u'(1) = 0\}$$ $$Au = u''.$$ Can someone explain how to calculate the adjoint of A, $A^*$, and the domain of $A^*$, ...
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33 views

Does this show that it is a bounded linear operator?

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$. I want to show that $\displaystyle e^{At}:=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}=T(t)$ defines a strongly continuous semigroup (i.e. a ...
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1answer
35 views

Calculating a norm of an operator

Let $T \in (C([a, b]))^*$, $$ T(u) = \underset{a}{\overset{(a+b)/2}\int} u(x) dx - \underset{(a+b)/2}{\overset{b}\int} u(x) dx. $$ Show that $ || T || = b - a $. We have that $$|| T || = ...
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1answer
27 views

When does the $L_1$ convergence imply almost everywhere convergence?

I know that $L_1$ convergence implies existence of an almost everywhere converging subsequence. But I was wondering, can you tell me some extra conditions on functions that make $L_1$ convergence ...
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1answer
31 views

Disjoint convex sets which cannot be separated by any continuous linear functional

This problem is out of Rudin's Functional analysis exercise 3.2. The problem is stated below. I'm really struggling with this chapter in general. It has a lot of new topics I have not seen before. Any ...
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1answer
34 views

How can I fix this proof using transfinite induction of the existence of bases of normed vector spaces?

I want to prove that every normed vector space has a basis. The following proof relies on the principle of transfinite induction. I believe that it is flawed because I'm not so sure if it's possible ...