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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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 ...
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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 ...
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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
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1answer
54 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
86 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 ...
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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
28 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 ...
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0answers
24 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
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1answer
51 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 ...
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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
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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 ...
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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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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 ...
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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
17 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} ...
0
votes
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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1answer
51 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
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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 ...
3
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2answers
48 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
1
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1answer
24 views

Spectral Measures: Boundedness

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: ...
1
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1answer
7 views

unitization of a Banach algebra

For any algebra $A$, the linear space $A_{1}=A+C=\{(a, k)|a \in A, k \in C\}$ equipped with the multiplication $(a,k)(b, l) = (ab+kb+la, kl)$, so-called the unitization of $A$, is a unital algebra ...
2
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1answer
38 views

A question in Hahn-Banach theorem

Let $X$ is a real vector space(without topology). call a point $x \in A \subset X$ an internal point of $A$ if $A-x$ is an absorbing set.Suppose $A$ and $B$ are disjoint convex set in $X$ and $A$ has ...
1
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0answers
21 views

countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
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0answers
10 views

Compactness in the weak topology generated by dual pair

Let $(X, Y)$ be a dual pair of normed spaces and $\sigma(X, Y), \sigma(Y, X)$ be weak topologies on $X, Y$ respectively. I would like to ask what are the conditions to guarantee $B_X$ compact with ...
0
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1answer
43 views

Normal Operators: Transform (III)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$N=Z\sqrt{1-Z^*Z}^{-1}$$ Especially one had: ...
0
votes
1answer
34 views

Spectral Measures: Invertibility

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: ...
0
votes
1answer
12 views

Gateaux derivative of a functional $f:\mathbb{R}^2 \to \mathbb{R}$ and why the Frechet derivative of it does not exist

I am more interested in the method which I use for it to be done. I need to understand all the in-between steps so that I can apply it to other examples too. Given the functional: \begin{equation} ...
0
votes
1answer
39 views

Question about norms $p$ and $q$.

I have a simply question: Show that if $x,y \in \mathbb{R^n}$, then $$\biggr|\sum{x_jy_j}\biggr|\leq \| x \|_p \| y \|_q$$ First, I proved that, if $s,t\geq 0$, then ...
1
vote
1answer
45 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
0
votes
1answer
32 views

Non-integer order derivative

I do not know much about fractional calculus, except what I have read in a few short posts at MSE and https://en.wikipedia.org/wiki/Fractional_calculus. I know that order of a derivative can be ...