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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27 views

Borel Measures: Coproduct

I need this thread as lemma! (See the advice: SE: Q&A) Given Borel spaces $\Omega_\lambda$. Consider the coproduct: ...
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1answer
16 views

Spectral Measures: Multi Version (II)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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13 views

Gelfand triple question

This is probably a stupid question, but given a triple $(\Phi,H,\Phi^*)$. If under such an example, if $\Phi$ is a Hilbert space and $\Phi$ can be identified with $\Phi^*$, why does this not mean ...
1
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1answer
22 views

Spectral Measures: Multi Version (I)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad ...
1
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0answers
22 views

Any relation between Kernel methods and Variational methods?

I am familiar with kernel method, which is defined in the link: https://en.wikipedia.org/wiki/Kernel_method On the other hand I am familiar with variational methods which is defined in the link: ...
1
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1answer
41 views

How to apply Cauchy's formula in proof 10.13 of Rudin's FA

Let $A$ be a Banach algebra and $x\in A$. In part of prood 10.13 of Rudin's Functional Analysis (page 254), where he is trying to prove that $$\rho(x) = \lim_{n\to\infty}||x^n||^{\frac{1}{n}} = ...
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36 views

Functional equation with shift operator

I'm trying to understand how can i apply Lagrange representation of shift operator and/or topology of $C^\infty$ to solve some sort of functional equations (and i'm not sure it's even possible). E.g. ...
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1answer
24 views

Leray-Schauder fixed point theorem and remark

Leray-Schauder fixed point theorem from Gilbarg and Trudinger book is quoted below. I do not understand remark below this theorem. Could you explain? Theorem 11.2 from this text is Schauder fixed ...
0
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1answer
14 views

Bounded linear space (elementary question)

Does exist (nonzero) bounded normed space over any field? Fix normed linear space $L$ over field K. We have $x+x+x+\cdots\in L$ So $||nx||=n||x||\rightarrow \infty$ when $n \rightarrow \infty \\$ ...
2
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0answers
61 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
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0answers
8 views

infinite compostion of functions: are there generic conditions when this can be done?

I came recently to think about the following problem. Imagine that the followig iteration law is given $$ x_k = f(x_{k-1},u_{k}) $$ If we iterate twice $$ x_k = f(f(x_{k-2},u_{k-1}),u_k) $$ or ...
2
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1answer
43 views

What is a integral of operators?

I am reading the book Semigroups of Linear Operatos and Applications to Partial Differential Equations which studies a uniformly continuous semigroup, this is a family $(T_t)_{t \geq 0}$ of bounded ...
0
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1answer
28 views

Upper semi-continuity proof for topological spaces

Hi does anyone have any idea or a possible hint for a proof of the following result: Consider asymmetric norm $p$ on $\mathbb{R}$ given by $p(t) = t^{+}$, for $t \in \mathbb{R}$. Show that if $(X, ...
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1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
0
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1answer
31 views

Find the norm of operator

Let we have the following operator $$T:L^p \rightarrow L^p$$ $T(ξ_1,ξ_2,ξ_3,ξ_4,ξ_5,ξ_6,ξ_7,ξ_8,ξ_9,ξ_{10},ξ_{11},ξ_{12},ξ_{13},ξ_{14},..)=(ξ_1,ξ_3,ξ_5,ξ_7,ξ_9,ξ_{11},ξ_{13},...)$ How can I find the ...
1
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1answer
18 views

How do I prove that there doesn't exist a unit norm vector at a unit distance from a closed subspace of an infinite dimensional vector space?

Let $M$ be a proper closed linear sub space of a normed linear space $X$. If $X$ is finite dimensional, it's a well known result by F.Riesz that there exists a unit vector $x$ such that ...
2
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1answer
52 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
3
votes
1answer
51 views

Positive-definite function and Positive-definite matrix

I am trying to understand Positive-definite function and read the wikipedia link: https://en.wikipedia.org/wiki/Positive-definite_function It has a relation to Positive-definite matrix and I did not ...
1
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0answers
29 views

Centralizer of $C^*$ algebra

Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra. If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for ...
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1answer
122 views

What would be the “action” in functional analysis?

I am reading Simmons' "Topology and Modern Analysis". He keeps bringing up the idea of studying the set of all structure preserving mappings to obtain info regarding the structure of a certain normed ...
3
votes
1answer
84 views

Is the space of almost everywhere differentiable function with bounded derivative embedded with uniform norm complete?

Let $A$ be the space of almost everywhere differentiable functions $[0,1]\rightarrow [0,1]$, and when differentiable, their derivatives are bounded by $M$. I'm aware that the space of almost ...
0
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1answer
26 views

Representation of a Functional Equation.

Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find ...
2
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0answers
50 views

Hahn-Banach from “Every vector space has basis”

What is the simplest way to prove Hahn-Banach starting from the AC-equivalent that every vector space admits a basis?
2
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0answers
49 views

Span of Polynomials in $\mathcal{C}(\mathbb{R})$ [duplicate]

Let $\mathcal{P}=\{1, x, x^2, x^3 \ldots\}$. Then pick out the correct statements. A) Span$\mathcal{P}=\mathcal{C}(\mathbb{R})$ B) Span$\mathcal{P}$ is a subspace of $\mathcal{C}(\mathbb{R})$ C) ...
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0answers
26 views

Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...
1
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0answers
23 views

Help in understanding Bochner's theorem and Pontryagin duality theorem

I am trying to understand Bochner's theorem through wikipedia link https://en.wikipedia.org/wiki/Bochner's_theorem This refers to dual spaces of locally compact abelian group and leads to ...
4
votes
1answer
50 views

Computing the norm $\|f\|$ of a functional.

Define $f: \ell^2(\mathbb{N}) \to \Bbb C$ by: $$f(x) = \sum_{n \geq 1}\frac{x_n}{n^2},$$where $x = (x_n)_{n \geq 1} \in \ell^2(\mathbb{N})$. It is pretty clear that $f$ is linear. Also, since ...
0
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1answer
33 views

Hilbert space isometric to a subspace of its dual

Let $\cal H$ be a Hilbert space, and let $\cal H^\ast$ be its dual (of the continuous functionals). If $\cal H$ is a real vector space, I can define: $$\begin{align}\Phi\colon\, &{\cal H} \to ...
1
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1answer
26 views

Spectral Measures: Adjoint

This thread is only Q&A! (See the hint: SE: Q&A) Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the ...
2
votes
1answer
20 views

Solving for the spectrum and eigenvectors of the “shift operator(?)” $T$ in $P_3(\mathbb{R})$?

This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$ (The actual question can be ...
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1answer
10 views

Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
2
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1answer
48 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
1
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0answers
24 views

Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
3
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1answer
37 views

Do the point-open and compact-open topologies coincide on $C([0,1], \mathbb{R})$?

Do the point-open and compact-open topologies coincide on the space of continuous functions from $[0, 1]$ to $\mathbb{R}$, i.e. on $C([0, 1], \mathbb{R})$? If not, what would be a clear and simple ...
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1answer
37 views

Describe the GNS construction [closed]

Question: Describe the GNS construction for the C$^*$-algebra $ C[0, 1]$ and for the positive linear functional $\phi $ given by $\phi(f) = f (0)$. What should i do? Should I describe Hilbert space ...
0
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1answer
57 views

Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$

Functions in D are finite test functions in $C^\infty(\mathbb{R})$ D' are distributions (genralized functions) Do I have to check that $\forall \phi \in D$: $\lim_{\epsilon \to 0} ...
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0answers
17 views

Total set in a Hilbert space

Definition: A subset of a Hilbert space is total if its span is the entire space. Halmos in his book (A Hilbert space problem book) asks below question: There exists a total set in a Hilbert ...
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3answers
55 views

Two normal operator that commutes

Suppose $N\in B(H)$ is normal, and $T\in B(H)$ is invertible. Prove that if $TNT^{-1}$ is normal then $N$ commutes with $T^*T$. I can not any idea to prove it, just I know ...
1
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1answer
16 views

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Let $a=\{a_i\}$ be an arbitrary sequence of complex numbers with finitely many non-zero terms. Consider the function $f(x)=\sum_ia_i g(x-i)$, where $g$ is a good function. Prove that for any $p\in ...
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0answers
20 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
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2answers
28 views

Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
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0answers
19 views

Let $T \in B(X,Y)$. if ${x_n} \to x$ weakly and $T$ is compact, why dose $\left\| {T{x_n} - Tx} \right\| \to 0$? [duplicate]

Let $T \in B(X,Y)$. if ${x_n} \to x$ weakly and $T$ is compact, why dose $\left\| {T{x_n} - Tx} \right\| \to 0$?
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0answers
33 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...
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votes
1answer
50 views

Functionnal analysis: Why $\langle AAx,x\rangle\underset{(*)}{\leq} (\|A\|+m)\langle Ax,x\rangle-\|A\|m ?$

Let $(X,\langle\cdot ,\cdot \rangle)$ an inner vector space and $A\in \mathcal L(X)$ symetric such that $A\geq 0$. I set $m=\inf\{\langle Ax,x\rangle \mid x\in X, \|x\|=1\}$ and thus $A-mI\geq 0$. ...
1
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2answers
41 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
2
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0answers
21 views

Dimension of a Hilbert space

Halmos in his book (A Hilbert space problem book) says, 1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. 2- Also he proves if orthogonal dimension of ...
4
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1answer
44 views

Adjoint of $L^{1}$ space

I have a question about $L^{p}$ spaces. Question: Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Let us consider $f \in L^{1}(X)$ satisfying the following property: \begin{align*} \forall ...
0
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0answers
11 views

Continuous function on the Skorohod space

I have a process $(X,Y)\in D([0,T],\mathbb{R}^2)$ where $D([0,T],\mathbb{R}^2)$ is the set of cadlag functions with Skorohod metric. Let $A=\{\sup_{t\in[0,T]}|X(t)-\int_0^tX(s)Y(s)ds|>\epsilon\}$ ...
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0answers
32 views

Frechet derivative of an operator

Let an operator $T:C[a,b]\to C[a,b]$ be defined as: \begin{equation} (Tu)(x)=\int_{a}^{b}K(x,t)f(t,u(t))dt \end{equation} where $K:[a,b]\times[a,b]\to \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to ...
-1
votes
1answer
32 views

Two-sided closed ideals of $C(X,M_2(\mathbb C))$

Let $X$ be compact and Hausdorff space. I know all closed ideals of $C(X)$. I want to substitute $\mathbb C$ by $M_2(\mathbb C)$. What can we say about two-sided closed ideals of $C(X,M_2(\mathbb ...