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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Reproducing kernel Hilbert space and finding orthonormal basis from kernel function.

I was trying to extract an orthonormal basis of a reproducing kernel Hilbert space from the expression of the kernel function. I know my answer for finite dimensional reproducing kernel Hilbert space. ...
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1answer
134 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
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35 views

Space where the separation theorem doesn't hold

I have read the proof of the next separation theorem: Let X b a normed space in R and A a convex and open set that contains 0. Let b be a point wich is not in A then there exists f in X* such that ...
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1answer
135 views

Relationship between different topologies of bounded operators on a Hilbert space

I am self-studying functional analysis. Given that $B(H)$ are the bounded operators on a Hilbert space, $H$. I would like to ask how to formally prove that the weak topology is weaker than the ...
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29 views

Why is the n-fold tensor product of S(R) dense S(Rn) [duplicate]

I need a reference request for the following fact: Every Schwartz function on $\mathbb{R}^n$ is the limit of a series of elements $$ (x_1, \dots, x_n) \rightarrow h_1(x_1) \cdots h_n(x_n),$$ where ...
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35 views

Set of critical points of polynomial: why finite

Let $P$ be a $C^\infty$ real constant coefficients polynomial defined on $\mathbb{R}^n$ and let $Z(P)$ be its set of critical values, that is $Z(P)=P(\{\xi\in\mathbb{R}^n: \nabla P (\xi)=0\})$. I read ...
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1answer
102 views

Lim inf with norm and weak convergence

The following is an real analysis qualifying exam problem that I cannot solve: Suppose $X$ is a Banach space and that $(x_n)$ converges weakly to $x$. Show that $\liminf ||x_n|| \geq ||x||$. Using ...
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1answer
55 views

Sesquilinear Form as two Linear Operators

We know that two Linear Operators give rise to a Sesquilinear Form in the sense: $$\operatorname{s}(x,y)=\langle\operatorname{S}x,\operatorname{T}y\rangle$$ What about the converse: Does every (not ...
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1answer
30 views

spectum of self adjoint operators

Let $H$ be an Hilbert space and $S = \displaystyle{ \sum_{i=1}^nS_i}$ where $S_i$ (i=1...n) is self adjoint with compact resolvent . is it true that the spectrum of $S$ is the sum of the spectra of ...
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1answer
47 views

convergence of product of sequences in $L^2$

Let $\displaystyle (f_n)_n \subset L^\infty ([0,1])$, $ \displaystyle (g_n)_n \subset L^2 ([0,1]) $ and $\displaystyle f \in L^\infty ([0,1]) $, $\displaystyle g \in L^2 ([0,1]) $ such that ...
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2answers
68 views

Calculating square roots of operators using power series for $\sqrt{1 - z}$

The following is a theorem from Reed & Simon's Methods of Modern Mathematical Physics, Volume I. Here, we are working in a complex Hilbert space $(\mathcal{H}, (\cdot, \cdot))$, and ...
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87 views

Applying Open Mapping Theorem

Let $X$ and $Y$ Banach spaces and $F: X \to Y$ a linear, continuous and surjective mapping. Show that if $K$ is a compact subset of $Y$ then there exists an $L$, a compact subset of $X$ such that ...
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0answers
45 views

Why locally compact and why Hausdorff in the definition of $C_0$

Define a set $C_0(\Omega)$ to be the set of continuous functions $f:\Omega \to \mathbb C$ such that for every $\varepsilon > 0$ the set $\{\omega \in \Omega | |f(\omega)|\ge \varepsilon \}$ is ...
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0answers
60 views

Versions of Trace Theorems

I have a quick question about the Trace Theorem. I have been using Evans book of Partial Differential Equations to study Sobolev Spaces. The Trace Theorem is given as "If $U$ is bounded and $\partial ...
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5answers
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What is the difference between a Hamel basis and a Schauder basis?

Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a ...
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1answer
65 views

Why does $e_i \in \ell^2$ weakly converge to $0$?

In our lecture notes of functional analysis we are given the following definition: Let $(X, \| \cdot \|_X)$ be a $\mathbb{R}$-vector space with a norm and $X^*$ be its dual space. Let $(x_k)_{k ...
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1answer
60 views

A question about extensions of Markov semigroups

I've cross-posted this to MO, if a reply appears on that post I'll update this one. Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on ...
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1answer
96 views

Innerproduct in space of holomorphic functions

I have a problem with the following exercise (6.4.5) in "Foundations of real analysis" by Avner Friedman: Let $D$ be the disc $|z|<1$ in the complex plane. Denote by $H^2(D)$ the linear subspace of ...
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2answers
233 views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function ...
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1answer
39 views

property of a given subset $ A $ of $\mathit{l}^2 $

I was given this exercise Let $\lbrace e_n \rbrace $ be the canonical basis of $\mathit {l}^2 $ (the space of square sommable series with the usual norm) and set $$ A:= \lbrace \sum_{n \ in ...
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1answer
60 views

Polarization Identities for Real Biliniear Forms

I'm tryin to find a derivation for polarization identities. I startet like this: Let $\operatorname{s}$ be a sesquilinear resp. bilinear form not necessary hermition resp. symmetric. Consider the ...
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2answers
166 views

2nd order partial differential equations

I have a question regarding based on text regarding 2nd-order partial differential equations. Consider the quasilinear 2nd-order partial differential equation: $$-\text{div}(a(x,y,\nabla u)) + ...
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0answers
61 views

Test functions on a compact interval

Consider a functional $E:C([0,1]) \rightarrow \mathbb{R}$ of the form $$E(g) = \int_0^1 g(s)ds$$ In dealing with such functionals one often needs test functions. If one talks about the space ...
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1answer
57 views

Show that if operator $T$ is such that $||I-T||<1$ , then $T$ is bijective.

I came across this statement in a proof and I can't figure out why its true, could someone point out why (or give a hint). Thanks. Suppose that $T:X\to X$ is a bounded linear operator that maps a ...
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45 views

Premetrics, where are they useful?

Wikipedia defines a premetric as a function $d : X\times X \to \mathbb R$ such that $d(x,y) \ge 0$ $d(x,x) = 0$. For me these axioms are so weak that I am wondering where they are used, do you ...
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1answer
87 views

Phillips spectral theorem

In Reed-Simon (see References) the following theorem due to Phillips is cited (but not proved): Theorem (Phillips). Let $X$ be a Banach space, $T \in \mathcal L(X)$. Then $\sigma(T) = \sigma(T')$ ...
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1answer
85 views

Dense subset in Hilbert space given by the span of an orthonormal basis

Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$. Let $X$ be the set of all finite sums $\sum\limits_{i=0}^n \lambda_i e_i$. Show that $X$ is dense by showing that $X^{\perp}=\{0\}$. Now ...
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1answer
30 views

Bounding an $L^2$ function (with Cubic…)

Suppose we have a function $f(x) \in H^1(R)$. Is it possible to bound $\int f^4 dx \leq \|f\|_{H^1(R)} ^3$. Likewise, it is possible to say $\int f^3 dx \leq \|f\|_{H^1(R)} ^3$? Thanks.
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1answer
204 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
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1answer
133 views

Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: $\quad$ Let $V$ be a Hilbert space, $U$ an open neighborhood of $u\in V$, and let $\varphi\in C^2(U,\mathbf R)$. Define implicity the linear operator ...
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2answers
108 views

$\|.\|_2$ closure of a set which is dense in $L^2[0,2\pi].$

The following is an exercise of Conway's Functional analysis, chapter 1, section 5. Let $L=\{f\in C[0,2\pi]|f(0)=f(2\pi)\}$ and show that $L$ is dense in $L^2[0,2\pi]$.
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46 views

Is it reasonable to think of the expectation of an infinite-dimensional vector?

Given a probability space $(\Omega, \mathcal{F}, P)$, a random vector is an $\mathcal{F}$-measurable mapping $X: \Omega \rightarrow \mathbb{R}^{k}: X(\omega) = (X_{1}(\omega), ...
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1answer
21 views

Verify that a Weak-neighborhood of a point is contained in the kernel of a continuous linear functional

I've encountered several times this reasoning but i can't find a good answer to it. Let $ E, F$ be banach spaces Given $\varphi : E \to F $ linear and continuous w.r.t the weak topology $\sigma (E, ...
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1answer
68 views

Riesz Representation Theorem for functions in $C_0(X)$

I've been analyzing the proof of Riesz Representation Theorem as presented in Rudin's book for Real and Complex Analysis as Theorem 6.19 and found some steps confusing. First I'd like to know why we ...
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1answer
61 views

Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
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1answer
48 views

clarification about theorem $3.10 $ of Brezis functional analysis book.

I'm referring to the theorem at page $61$. It shows that for a linear operator $T $ between $E $ and $ F$ Banach space are equivalent (notation: $S$ means strong topology, the norm ome, $W $ weak ...
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2answers
232 views

Definition of Banach limit

In my Bachelor Thesis I have defined a Banach limit as a functional $LIM: l^\infty (\mathbb{N})\rightarrow \mathbb R$ that has the following properties: B1 If $(x_n)$ is a convergent sequence, then ...
13
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1answer
197 views

Is This Set of Zero Measure?

Let $(X,\mathscr M,\mu)$ be a measure space and $(Y,\|\cdot\|)$ a separable Banach space with $\{y_n\}_{n=1}^{\infty}$ being a dense subset in it. Suppose that $f:X\to Y$ is a Borel measurable ...
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1answer
133 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
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1answer
56 views

Baire category related question

Let $A_n$ be a sequence of closed sets of $\mathbb{R}$ such that $[a,b]\subseteq \cup A_n$, for some $a<b $. Show that at least one of the set $A_n$'s contains an interval. Thanks in advance,
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1answer
48 views

Homeomorphism between space of characters and compact set

I am not sure how to construct homeomorphism between $\Omega(C^1([0,1]))$and $[0,1]$, where $\Omega(C^1([0,1]))$ denotes set of characters over algebra $C^1([0,1]).$ I know that homeomorphism between ...
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1answer
70 views

complement of a dense set

Corollarie (b) after theorem 4.12 of Rudin's Functional Analysis states $\mathscr{R}(T)$ is dense in $Y$ if and only if $T^*$ is one-one. Where $T:X \rightarrow Y$ is a bounded linear operator ...
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1answer
59 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
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0answers
68 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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1answer
150 views

convergence in weak Lp

I've seen this statement so I'm not wanting to argue it, but am thus curious as to where my logic falls short. Folland 6.2 problem If $1<p<\infty, \;f_n\rightarrow f$ weakly in $l^p(A)$ iff ...
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1answer
46 views

Sobolev spaces inclusion

I'm having trouble finding an answer related to Sobolev spaces that does not relate to duality. I'm looking for an answer to the following question: When (i.e. for what domains $\Omega$ or such) can ...
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1answer
37 views

$T\in B(H)$ normal and left invertible implies $T$ invertible?

My question is what's written in the title, that is, if $T$ is a normal operator on a Hilbert space $H$, and $T$ is left invertible, is it necessarily true that $T$ is invertible? Actually, the more ...
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1answer
129 views

Linear Operator and isomorphism

I wanted to be sure about the following: Let's say we have vector spaces normed spaces $X$ and $Y$ and a linear operator $T:X \rightarrow Y$. My idea was to reduce the properties that I need to show ...
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1answer
37 views

Dominated convergence theorem query

Why is the hypothesis that the sequence $(f_{n})$ is dominated required? If $f_{n} \to f$ pointwise, then $f_{n}-f$ is bounded and tends to 0 so applying the theorem to $f_{n}-f$ gives the desired ...
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votes
3answers
212 views

Is the sphere compact?

Riesz' lemma gives us that in infinite-dimensional spaces no ball is compact. but what is about the sphere$=\{x \in X; ||x||=1\}$? can we say something about the compactness of the sphere in ...