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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339 views

Proving two results about the spectral radius

How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces? Theorem 1. Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon ...
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vote
1answer
93 views

Characterization of strong minimums with slices.

I am doing a proof of a Lemma that isn't in a book. Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$. I have that ...
5
votes
1answer
197 views

The ratio of two $L^p$ norms

Let $f$ be a non-negative function on a measure space $(X,\mu)$ with $\mu(X) = 1$. Is there a known characterization of when $$\lim_{n \rightarrow \infty } \frac{\|f^n\|_p}{\|f^n\|_q } = 1,$$ for ...
2
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2answers
209 views

Showing dual space of a normed space is a normed space

Let $X$ be a Normed Linear Space. The dual space $X^*$ of $X$ is the set of all bounded linear functionals on $X$. It is a normed linear space with the norm $\Vert\varphi\Vert$. (according to notes). ...
5
votes
1answer
155 views

Generalized notions of mixture

A subset $S$ of a real vector space is convex if it is closed under finite mixtures: for any $\lambda_1,\ldots,\lambda_n>0$ such that $\lambda_1+\cdots+\lambda_n=1$ and any $x_1,\ldots,x_n\in S$, ...
2
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0answers
263 views

Concept of Hilbert triple

I am trying to understand the Hilbert triple $V \subset H \subset V^*$, where $V$ and $H$ are Hilbert spaces and the star denotes the dual space. Eg: $H^1 \subset L^2 \subset H^{-1}.$ If $V \subset ...
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votes
0answers
97 views

The spectrum of Schrodinger operator

Assume the Schrodinger operator $-\Delta+u$ is self-adjoint($u\to 0,u_x\to 0$, when $x\to \infty$),so the spectrum of it is real. I can prove that the point spectrum $\lambda<0$,but I can't prove ...
17
votes
1answer
1k views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
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1answer
441 views

Proving that an operator is unbounded and not closed.

I am revising for my Functional Analysis exam and am stuck on the following question. Consider the Hilbert Space $L^2([0,1]),$ and define the operator $T:D(T) \rightarrow L^2([0,1])$ by ...
6
votes
4answers
956 views

Please suggest a functional analysis book to refresh my knowledge

I would like to ask you to recommend me a good modern textbook on functional analysis to refresh what I already know. I am a computer science student and for the last two semesters we've been having a ...
0
votes
1answer
112 views

asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ...
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0answers
160 views

Is there a deeper connection between the two Riesz's Representation Theorems?

I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems: (1) Every bounded linear functional $f$ on a Hilbert space $H$ can be ...
2
votes
1answer
158 views

Principal Valued Distributions

I am currently studying applied functional analysis and I see a proof about principal valued distributions. It is easy to prove that $x \times P/x =1$, where $P/x$ is the principal valued ...
3
votes
1answer
202 views

Strongly-Continuous linear functionals on $\mathcal{B}(H)$

Suppose $H$ is a complex Hilbert space and $$w: \mathcal{B}(H) \longrightarrow \mathbb{C}$$ is a bounded linear functional on $\mathcal{B}(H)$ such that $w$ is continuous even if $\mathcal{B}(H)$ is ...
8
votes
1answer
590 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
0
votes
2answers
182 views

In a C*-algebra, does $a \leq b$ imply $a^2 \leq b^2$?

While attempting to fill in the gaps in a proof of the Gelfand-Naimark-Segal representation theorem that I was given in a course in operator algebras, I found myself wondering whether, if ...
2
votes
1answer
190 views

Operator Norm $\| T\|$

Would somebody mind explaining why if $T$ is a continuous and bounded operator on a Hilbert space $H$, we have $$\|T\| = 1 \;\;\;\Rightarrow \;\;\;\|Te_n \| = \|e_n\|\;\;\mbox{for all }\;\;x\in H$$ ...
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0answers
50 views

existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
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0answers
225 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
6
votes
1answer
451 views

Finite dimensional subspaces

Let $X$ be a complex Banach space of infinite dimension. Does there exist a finite dimensional subspace of $X$ of arbitrary (finite) dimension which is complemented by a projection of norm 1?
7
votes
1answer
272 views

Do the two limits coincide?

Let $a$ be a non negative (positive almost everywhere) weight in $L_{loc}^1(\Omega)$, $\Omega\subseteq\mathbb{R}^n$ is open. For $\varphi\in C_c^{\infty}(\Omega)$ define $$ ...
3
votes
1answer
213 views

Spectral radii and norms of similar elements in a C*-algebra: $\|bab^{-1}\|<1$ if $b=(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$

Let $A$ be a unital $C^*$-algebra and $a\in A$ such that $r(a) < 1$. Define b = $(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$. We can prove that $b\geq e$ and that $b$ is invertible. I want to show $\| ...
6
votes
1answer
110 views

Bounded extension

What are the easiest examples of a pairs of Banach spaces $X,Y$ such that $X\subseteq Y$ ($X$ is a closed linear subspace of $Y$) there is a bounded linear map $T\colon X\to Y$; there is no bounded ...
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0answers
89 views

Total sets in $R$ compared with total sets in $[a,b]$

A total set in a NLS is one whose linear span is dense in the set. e.g. $A = \{1,x, x^2,...\}$ is total in $(C[a,b],\Vert\cdot\Vert_{\infty})$ I find it easier to talk about total sets than dense ...
2
votes
1answer
76 views

asymptotic limit at the integral

I would like to get an asymptotic limit at the following integral: for $p\ge 2, n \in N$, $t \ge 0$ $$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$ I think ...
4
votes
1answer
140 views

Prove that this property holds for any $f\in L^\infty([0,1])$.

I recently came across this problem, namely we are given a continuous function $f:\mathbb R\to\mathbb R$ such that $$\int_0^1f(u(x))\mathrm dx=0,\;\forall u\in C^0([0,1]):\int_0^1u(x)\mathrm d ...
4
votes
1answer
366 views

Maximize the integral of $f$, knowing the integral of $\frac{1}{f}$, for a Lipschitz function $f$

This question is related to this recent other question, where two intervals $[a,b] $ and $[c,d]$ were considered. Here I ask about a simpler version with just one interval $[a,b]$. Consider the ...
4
votes
1answer
254 views

How to prove that there is a subspace $W \subset C(X)$ so that $C(X)$ is isomorphic?

Let $X$ be a compact metric space and let $F$ be a closed subset of $X$. Assume that there exists a bounded extension operator $T:C(F) \rightarrow C(X)$, i.e., $T \in B(C(F),C(X))$ and for all $g\in ...
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vote
1answer
322 views

Compact operator? self adjoint operator? Stirling's formula

Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $M\colon\ell^2 \to\ell^2$ as follows: $$S(x_1,x_2,x_3,x_4,\ldots)=(0,x_1,x_2,x_3,\ldots) $$ and ...
6
votes
2answers
274 views

Completeness and Fourier series convergence

Consider the question: In an inner product space $V$, when does the Fourier series of $x$, $\sum\limits_{n=1}^k\langle e_n,x\rangle e_n$ converges to $x$ as $k\to\infty$? Well, certainly is converges ...
3
votes
1answer
519 views

For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.

I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
1
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1answer
307 views

$L^2([0,1])$ is a set of first category in $L^1([0,1])$?

How to show that $L^2([0,1])$ is a set of first category in $L^1([0,1])$? Thank you.
4
votes
1answer
156 views

$T(V)$ is a closed subspace of $V$?

Let $V$ be a normed vector space (not necessarily a Banach space) and let $S$ and $T$ be continuous linear transformations from $V$ to $V$. If we assume that $T=T \circ S \circ T$. Then how to show ...
3
votes
1answer
101 views

How to prove here exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite.

How to show that let $(X,\mathcal{M},\mu)$ be a measurable space, there exists a function $f \in L^1(X,\mu)$ with $f>0$ $\mu$-a.e. iff $\mu$ is $\sigma$-finite. Can you please help me out? Thank ...
2
votes
0answers
232 views

Representation of compactly supported distribution

Is this true? Any compactly supported distribution $T\in \cal D'$ can be represented as finite sum of partial derivatives of functions.
1
vote
1answer
235 views

Reference for studying the method of stationary phase

I would like to learn about the stationary phase method, as part of the theory of Fourier Integral Operators. From what I can see so far, Hormander's book "The Analysis of Linear Partial Differential ...
11
votes
1answer
513 views

Elementary applications of Krein-Milman

Recall that the Krein-Milman theorem asserts that a compact convex set in a LCTVS is the closed convex hull of its extreme points. This has lots of applications to areas of mathematics that use ...
2
votes
2answers
164 views

Is $L^2(D)$ separable?

Let $D$ be a bounded connected open subset of $R^n$ and $μ$ is a finite measure on $D$, say the Lebesgue measure. Is $L_2(μ)$ separable? Is a bounded sequence $\{f_k\}$ of $L^2(μ)$ pre-compact?
4
votes
1answer
347 views

what exactly is weak* topology?

I know that weak* topology is the weakest topology so that $Jx$ is continuous for $\forall x\in X$, where $J$ is the isometry from $X$ to $X''$. But what exactly is this topology? What is the open set ...
4
votes
1answer
429 views

Weak convergence

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
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vote
2answers
94 views

weak convergence on $\ell_1$

Let $\{x^k\}$ be a weak convergent sequence in $\ell_1$, and its weak limit is 0. Is the following property true: For $\forall \epsilon >0$ and $\forall n>0$, there exists a K, s.t. ...
4
votes
1answer
317 views

solution for the degenerate parabolic PDE

Look at $u_t=a(x)u_{xx}$ if I have $a(x)\geq a_0>0$ then I can see in all books that $C^{2,1}$ solution exist and it is unique. However, if $a(x)\geq0$, that is degenerate, I see in Friedman's book ...
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0answers
79 views

tensor product, notation, and the sup norm

This question is a bit tricky for me to post because I don't really understand all the symbols and techniques involved - so it is likely that I'll do mistakes. In case it is unclear what I write ...
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votes
0answers
662 views

$L^2$ norm and its discrete analogy

$L^2$ is space of functions with $||e||^2$ as usual integral, so it is infinite dimensional, so there is no norm equivalence rule holds there. the discrete analog of its norm is $||e||^2_2=h\sum{_1^m ...
7
votes
2answers
211 views

Set operations in the constructions of the Weak Topology cannot be reversed

Let $X$ be a generic set and let $(Y_i)_i$ be a family of topological spaces. Let $(\varphi_i)_i$ be a collection of functions of the kind $X \to Y_i$. It is possible to determine a topology (that ...
2
votes
1answer
127 views

Linear image of closed convex subset

Let $X$ and $Y$ be two Banach spaces and $A:X\rightarrow Y$ a linear, continuous map. Let $M\subseteq X$ be a closed, convex subset of the unit sphere in $X$. When is $A(M)\subseteq Y$ closed?
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votes
2answers
343 views

Examples of not completely bounded maps

Let $\phi:\mathcal{A}\longrightarrow\mathcal{B}$ be a bounded map between $C^*$ algebras. $\phi$ is said to be completely bounded if the natural extension map \begin{eqnarray} ...
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vote
2answers
334 views

Hilbert space operators, relation between trace, rank and range

If $A\colon H\to S$ is a bounded operator on a Hilbert space $H$, and $S\subset H$. It is known that $\operatorname{trace}(A)=\sum_{n} \langle Af_n,f_n\rangle$ for any orthonormal basis $\{f_{n}\}$. ...
1
vote
1answer
252 views

uniform convergence of continuous functions

Let $X=C[0,1]$ and $x,x_{n}\in X,\forall n\in\mathbb{N}.$ Suppose $\forall t\in[0,1],\ x_{n}(t)\rightarrow x(t)$ and $\sup_{n}\left\Vert x_{n}\right\Vert <\infty$. Show that there exist convex ...
5
votes
2answers
173 views

$L^{2}$ functions

Let $f(x)$ be a continuous function for all $x\in \mathbb R$, such that $f\in L^{2}(\mathbb R)$ (i.e., $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$), and define $$f_{o}(x):=\sup_{|x-y|\leq ...