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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Absolute Convergence of a Series defined as a Cauchy Sequence

So the question I'm answering is "Suppose (X, || ||) is a normed space. Show that X is complete iff every absolutely convergent series in X converges on an element of X." The first half was simple (...
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59 views

Why should the map $-\Delta^{-1}$ continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
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33 views

Two equivalent statements for a sequence of $L^p$ functions

This problem showed up on a previous qualifying exam and I have had issues trying to solve it. Let $(X,\cal{M},\mu)$ be a positive measure space. Let $p,q\in(1,\infty)$ satisfy $\frac{1}{p}+\frac{...
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1answer
70 views

Intuition for orthogonality in infinite dimensions

I'm trying to explain orthogonality in inner product function spaces (e.g. Hilbert spaces) intuitively. As main expample, take the $L^2$ inner product given by $$<f,g>_{L^2(I)}:=\int_I f(x)g(x)...
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102 views

Neighborhood base at zero in a topological vector space

I'm reading a proof of a theorem which characterizes the collections of sets which can serve as a neighborhood base at zero in a topological vector space: Here are my questions: Why "This shows ...
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80 views

Uniform norm $ \|u\|_{C(\overline{U})}$ in PDE

Let $U\subset \Bbb{R}^n\to\Bbb{R}$ be an open set (not necessarily bounded) and $u:U\to\Bbb{R}$ be a bounded continuous function. In Evans's PDE textbook, the author defines a norm $$ \|u\|_{C(\...
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L^p space question [duplicate]

I am just wondering whether following is true? Let $p>1$ If $f_n\to f$ in $L^p(\mu)$, is it then true that $f^p\to f^p$ in $L^1(\mu)$? This is true if $||x|^p-|y|^p|\le |x-y|^p$. I know this is ...
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56 views

Strongly continuous semigroup

Given the following PDE $$\left\{\begin{array}{lll}u'(t)&=&A(u(t)) + f(t)\ \mathrm{for\ all}\ t \geq 0\ ,\\ u(0)&=&u_0\\\end{array}\right.$$ with $D(A) = H^2(0,1) \cap H_0^1((0,1))$ ...
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94 views

Is this proof of convergence in $\ell_p$ vaild?

Let $\{x_{n,i} \}$ be a Cauchey seq. of sequences in $(\ell^{p},d_{p})$ , $ \ell^{p}= \{ \{x_{n} \} ; \sum_{n=1}^{\infty}\mid {x_n} \mid ^{p} < \infty \} $. Show \begin{equation*} \lim_i \sum_{...
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129 views

F. & M. Riesz theorem

Can someone explain me in which sense F. & M. Riesz theorem (https://en.wikipedia.org/wiki/F._and_M._Riesz_theorem) is important/interesting?
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74 views

$\displaystyle \lim_{p \to \infty} \lVert x \rVert _p = \lVert x \rVert _{\infty}$ is uniform in $x$?

There is some sense in saying that the limit of $\ell^p$ norms $\displaystyle \lim_{p \to \infty} \lVert x \rVert _p = \lVert x \rVert _{\infty}$ is uniform in $x$? Maybe with this definition $(\...
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2answers
112 views

What does the notation $C(\bar U)$ mean for $U\subset\Bbb{R}^d$ open?

Let $U$ be an open subset of $\Bbb{R}^d$. In Evans's PDE book, $$ C(U)=\{u: U\to\Bbb{R} \mid u\ \hbox{continuous}\} $$ and $$ C(\bar U)=\{u\in C(U)\mid u\ \hbox{ is uniformly continuous on bounded ...
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30 views

How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space $(X,\mu)$....
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55 views

$H$ is Hilbert, countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle \forall y\in H$. Show $||x_n-x|| \to 0$

Problem Statement: Suppose $H$ is Hilbert, with a countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle$ for all $y\in H$. Show $||x_n-x|| \to 0$. My attempt: I'm ...
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71 views

Does a positive homogeneous and subadditive functional have to be nonnegative?

Do positive homogeneity and subadditivtity imply nonnegativity?
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70 views

Boundedness of smooth functions approximating an Lp function

We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f ...
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57 views

Difficulty in understanding converse part of proof of a propostion in Andrew Browder's Mathematical Analysis

Proposition: Let $\mu$ be finitely additive set function, defined on the algebra $\mathscr A$. Then $\mu$ is countably additive if and only if its has following property: if $A_n \in \mathscr A$ and $...
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1answer
21 views

checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any $y\...
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84 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
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1answer
36 views

Unitary elements in Banach spaces and subspaces.

Let $F$ be a Banach space and $E$ be a subspace of $F$. Let $e_{0}\in E$ be an element of norm $ 1$ and suppose that span $\{f\in F^{*}:\|f\|=f(e_{0})=1\}=F^{*}$, where $F^{*}$ is the dual space of $...
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42 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
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31 views

About closedness and boundedness of $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$

Let $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$. To check which one is true: (a) $H$ is bounded (b) $H$ is closed (c) $H$ is a subspace (d) $H$ has interior points My ...
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1answer
108 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
2
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1answer
105 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
2
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1answer
38 views

A discontinuous function weakly converging.

This is an exercise from a course. Suppose that $f \in C^{\infty}_{C}(R^3),$ and consider the function for t $\in [0,1)$, $$v(t,x)= \frac{1}{1-t}f(\frac{t}{1-t}).$$ Show that $v \in L^{\infty}([0,1),...
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99 views

The minimizer of a bounded continuous function over a closed set lies within the set?

Let K be a closed subset of a Hilbert space X. I was trying to prove that the infimum of the distance between K and any point x in X $$\inf_{y \in K} ||x-y||$$ is achieved by a point in K itself. The ...
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40 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that $g:\mathbb{R}^2\...
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71 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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1answer
55 views

Closed subsets of Banach space

$X=\{(a,b)\mid a \in C[0,1],b \in C[0,1]\}$, and its norm is $\|(a,b)\|=\|a\|_\infty+\|b\|_\infty.$ $Y=\{(a,a')\mid a \in C^1[0,1], \ a'(t)=\frac{da}{dt} \},\ Z=\{(0,b)\mid b \in C[0,1]\}.$ ...
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148 views

Spectrum of weighted shift operator

The Banach space considered is the following: $(l^{\infty}(\mathbb{Z}), \|\cdot\|_{*})$ with $\|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|$. Define $A$, an ...
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1answer
80 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
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62 views

Help understanding Rudin's proof showing that $C_c(X)$ is dense in $L^p(\mu)$

The proof is from Rudin's "Real and Complex Analysis." It states For $1\leq p<\infty$, $C_c(X)$ is dense in $L^p(\mu)$ The proof is Let $S$ be the class of all complex, measurable, simple ...
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52 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
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2answers
58 views

How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
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1answer
64 views

Normal Operators: Von-Neumann

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}N\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Regard their algebra: $$\mathcal{A}(N):=\{\eta(N):\eta\in\mathcal{...
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1answer
31 views

How does this follow from the theorem?[normed linear space]

I have this theorem: Let X and Y be normed linear spaces and let $T:X\rightarrow Y$ be a linear transformation. The following are equivalent: a. T is uniformly continuous. b. T is ...
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1answer
42 views

Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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51 views

Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
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172 views

Polar Decomposition: Adjoint

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\subseteq\mathcal{H}\to\mathcal{K}:\quad A=A^{**}$$ Polar decompose: $$A=J|A|:\quad J^*J=...
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120 views

Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
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1answer
35 views

A question about Chapman-Kolmogorov equation

I'm reading ''Functional Analysis'' - K. Yosida and at page 379 there is the following claim "The hypothesis that the particle has no memory of the past implies that the transition probability P ...
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1answer
74 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ‎‎...
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1answer
44 views

Convergence of a sequence of subspaces

Let $E_n\subset \mathbb R^n$ be a sequence of subspaces. What does it mean $E_n$ convergence to a subspace $E\subset \mathbb R^n$? I saw this when reading about hyperbolic sets. Where can I read ...
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2answers
36 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
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1answer
87 views

A condition for surjectivity of a linear map

Let $V,W$ be vector spaces (not necessarily finite dimensional!), and let $W^*$ the dual of $W$. Let $$A:V\longrightarrow W^*$$ be a linear map. What conditions do I have to put on $V$ and especially $...
2
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1answer
89 views

Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
2
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1answer
155 views

Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
2
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2answers
32 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
2
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1answer
122 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
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1answer
39 views

The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...