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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29 views

Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
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29 views

Can one prove the existence of a fixed point for a shrinking map on a sequentially compact metric space WITHOUT proving the space is compact?

Let $(X,\rho)$ be a metric space with $Y\subset X$ a sequentially compact subspace, and a mapping $T:X\to Y$ satisfying $\rho(Tx, Ty)<\rho(x,y)$ for all $x\neq y$. Prove that $T$ has a unique ...
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2answers
27 views

Showing interior of a set is empty

Consider the metric space $(C[0,1],d_{\infty})$. For $x_{0} \in [0,1]$ and $M > 0$ define the set $A \subset C[0,1]$ by $$ A = \{f \in C[0,1] \phantom{.}|\phantom{.} |f(x) - f(x_{0})| \leqslant ...
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13 views

About hyperplane separation theorem

I read in Lovasz's notes about semidefinite programs and combinatoric optimization. If $x_1A_1 + ... + x_nA_n\succ 0$ has no solution, then the linear subspace $L = x_1A_1 + ... + x_nA_n$ is ...
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2answers
24 views

Does weak convergence of $u_m(t)$ in $L^2(0,T;X)$ imply weak convergence of a subsequence of $u_m(t_0)$ in $X$ for a.e. $t_0$ in $[0,T]$?

In a book I'm reading (Navier Stokes Equations, by Constantin and Foias), the authors construct a sequence $u_m$ of functions in $L^2(0,T;V)$ which converge weakly to $u$ in $L^2(0,T;V)$. They then ...
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1answer
55 views

What's the precise mathematical theorem that's being cited when people write “if we let m go to infinity”?

I see this sentence used a lot but many times I'm confused by it. I imagine there's a theorem that's being implicitly cited every time this sentence is used. Could somebody please tell me which one it ...
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1answer
22 views

‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Let ‎$‎‎A$ ‎be a unital ‎‎‎$‎‎C^*$-algebra. ‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎ ‎let $‎‎C^*({a}) $ be the ...
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1answer
20 views

Form of the Polar decomposition for $M_{\varphi}$

Polar Decomposition:Let ‎$‎‎v$ ‎be a‎ ‎continuous ‎linear ‎operator ‎on a‎ ‎Hilbert ‎space ‎‎$‎‎H$.then ‎there ‎is a‎ ‎uniqe ‎partial ‎isometry ‎‎$‎‎u\in B(H)$ ‎such ‎‎$‎‎v=u‎‎\mid ‎v‎\mid‎‎‎$ ‎and ...
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8 views

Does it make sense to define continuity and monotony on Monte-Carlo simulations?

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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0answers
29 views

Why can we calculate the supremum of operator norm over unit circle?

I know that to check whether a linear operator is continuous or not we have to check if the operator norm is bounded. $$T: V\to W$$, $$\vert\vert \ T \vert\vert= \sup_{f \in V}\frac{\vert\vert \ Tf ...
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35 views

banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
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1answer
16 views

What does $u_0(x)$ represent?

I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$. what does this mean? What does $u_0(x)$ represent?
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70 views

Fermat like equation for meromorphic functions.

I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints? For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ ...
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1answer
9 views

Where does $u(t,x) \to u(t,x)-a-(b-a)x$ come from?

I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$ I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$. We assume ...
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15 views

Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
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1answer
23 views

Trouble understand a step in the proof that $l^p$ is complete

I'm reading through a proof, attached here. I didn't include the whole proof. The last step is the one I'm confused about. Shouldn't there be more of a justication for taking $\lim_{n \to \infty}$ ...
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23 views

Tools to study level sets of Lebesgue function

Given $u\in L^\infty(\mathbb{R}^N)$ with compact support, are there nice tools to study the level sets $$K: = \{x\in \mathbb{R}^N : u(x) = \|u\|_\infty \text{ a.e.}\}.$$ The main problem is $K$ being ...
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37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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2answers
39 views

Product spaces and open sets

I have a proposition I have been pondering that I need help with. Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Recall that the product space $(X\times Y, d_{1})$ is also a metric space with the ...
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1answer
77 views

Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
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1answer
51 views

Continuous but not compact operator on $L^2(0,\infty)$

Define the following operator on $L^2(0,\infty)$: $$Tf(x)=\frac{1}{x} \int_0^xf(y)dy,\quad f\in L^2(0\infty).$$ I would like to see that it is continuous but not compact. So, this is an integral ...
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1answer
36 views

Why the set of pure state ‎is ‎weak* ‎compact?

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎. ‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$. ‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ...
2
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2answers
28 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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2answers
19 views

Self-adjoint operators, projections, and resolutions of the identity.

In my Functional Analysis course, we're discussing the Spectral Theorem and the like. One question from a previous exam states the following: Let $H$ be Hilbert over $\mathbb C$, let $T \in B(H)$ be ...
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1answer
22 views

Closed ideals in $\mathbb B(H)$

Let $\mathbb{H}$ be a non-separable Hilbert space. If $\alpha$ is an countably many infinite cardinal number, let $I_{\alpha}=\{A\in \mathbb{B(H)}\:dim~ cl(ran A)\le \alpha\}.$ Show that ...
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18 views

Superset of the spectrum of the sum of a self adjoint operator and a bounded operator.

I'm stuck with the following problem: Suppose $A$ is a self-adjoint operator and $\Vert B-z_0\Vert\leq r$. Show that $\sigma(A+B)\subset \sigma(A)+\overline{B_r(z_0)}$, where $B_r(z_0)$ is the ...
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0answers
16 views

Dual space and norm to $X=c_0\times l^1$, solution check

$$X=c_0\times l^1,\ \|(x,y)\|=\|x^\|_{\infty}+\|y\|_1$$ It is clear, that the dual space is isomorphic to $l^1\times l^\infty$ and the functional $x^*(x)$ is defined as ...
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20 views

Neumann Series for integral equation with inhomogeneous term zero

Consider the method described in the following article: http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html In this notation, what happens when $ f(x)=0 $? All the terms seem to be zero ...
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1answer
217 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
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28 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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3answers
7k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
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5answers
2k views

The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
4
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2answers
48 views

Arzela-Ascoli for $\mathbb R^n$ from the case of $\mathbb R$?

In class, we proved the Arzela-Ascoli theorem for $\mathbb R$. The lecturer said it's also true for $\mathbb R^n$, and this version is deducible from $\mathbb R$. I tried to do this but failed. How ...
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1answer
65 views

Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
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2answers
175 views

Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
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1answer
17 views

Positive self-adjoint operators and norm resolvent convergence

I recently came across a reference to the following Theorem (Simon/Reed, Methods of Modern Mathematical Physics, viii.25) and am now trying to figure out a proof for it: If $A_n$ and $A$ are positive ...
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1answer
50 views

Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = ...
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41 views

Linear operator on $L^p$

Let $\{E_1, E_2, ..., E_n\}$ be a collection of pairwise disjoint measurable subset of $(\mathbb{R}, m)$ ($m =$ Lebesgue measure on real line) with $0 < m(E_i) < \infty.$ Then for each $1 \leq p ...
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1answer
47 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
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1answer
42 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
3
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1answer
29 views

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
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0answers
21 views

When are Fourier spaces included in each other?

Given the Fourier spaces $V(N_1, T_1)$ and $V(N_2,T_2)$, what necessary and sufficient conditions are required in order to have $V(N_1, T_1)\subset V(N_2,T_2)$? I know that if $V(N_1, T_1)$ is ...
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0answers
13 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
0
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2answers
45 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
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2answers
114 views

Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by \begin{equation} f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } ...
3
votes
1answer
2k views

Inner product is jointly continuous

I'm attempting another exercise from my notes: Show that an inner product on an inner product space is jointly continuous with respect to the induced norm:if $v_n \to v$ and $w_n \to w$ as $n \to ...
1
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1answer
20 views

Prove order of distribution $\Lambda_{1/x}$ is 1

The distribution is defined as: $$\Lambda_{1/x}(\varphi)=\lim_{\varepsilon\rightarrow0+}\int_{\mathbb{R}\backslash(-\varepsilon,\varepsilon)}\frac{\varphi(x)}{x}\ \mathrm{d}x$$ I tried integrating ...
1
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1answer
20 views

Largest invariant subspace

If $A$ is a $n\times n$ matrix with complex entries, denote by $inv(A)$ the dimension of a largest dimensional non-trivial invariant subspace of $A$. What is: $$\inf\{inv(A): ...
0
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2answers
32 views

Reflexive Banach spaces, compactness

Let $X$ be a reflexive Banach space. Then, consider a linear and compact operator $T \colon X \to X$. Prove that if: $\text{inf} \{ \|Tx\| : x \in X\quad \text{s.t.}\quad \|x\| = 1 \} > 0$, ...
2
votes
1answer
20 views

Small perturbations and eigenvalues

Suppose $A$ is a $n\times n$ matrix. Given $\epsilon>0$, can one find a rank one matrix $B$ with euclidean norm at most $\epsilon$ such that $A+B$ has $n$ distinct complex eigenvalues? Given a ...