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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15 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
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21 views

Random variables with all moments. Is this statement true?

Let $X$ be a random variable such thta $X\neq 0$, $P$-a.s. Then $$X\in \bigcap_{p\geq 1} L^p(\Omega) \iff \frac{1}{X} \bigcap_{p\geq 1} L^p(\Omega).$$ In other words, is the space $\bigcap_{p\geq 1} ...
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27 views

Determinant of solution of linear equation

Is there a direct way or method to know if the solution to a linear ODE is invertible? I mean, let $A(t)$ be a ($n$ times $n$) matrix and denote by $X(t)$ an unknown Matrix (of the same dimensions) ...
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46 views

Can a continuous linear form have a norm of infinity?

We know that a linear form $A \in V^{*}$ is continuous iff $$ \exists C: C\in R, ||Av|| \leq c ||v|| \forall v \in V $$ but we know too that $$ ||A|| = min\{c>0:||Av|| \leq C||v|| \forall v\in ...
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37 views

Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
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1answer
46 views

Image of $C^\ast$-algebra is closed?

Let $A$ be a non-zero commutative $C^\ast$ algebra and let $\varphi : A \to B$ be a homomorphism of star algebras. Please could someone help me how to show that $\varphi(A)$ is closed in $B$?
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33 views

Proving a Sobolev-type ineqauality

Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold: $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq ...
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32 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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1answer
29 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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91 views

$\langle Tx,x\rangle =0$ , then T is zero

I just wanted to be sure about something. The implication $\langle Tx,x\rangle =0$ , then T is zero , holds only if $T$ is self-adjoint right? If $T$ is an arbitrary operator, we need to have $\langle ...
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1answer
56 views

Prove that the union of two bounded set in a metric space is bounded.

Two bounded sets have finite diameter ..but how I prove that union of these sets have finite diameter to show union is bounded.
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56 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
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1answer
31 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...
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28 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
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1answer
64 views

The set of analytic functions on unit circle is not a C*-algebra

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed ...
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1answer
37 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...
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46 views

Is this map surjective?

Let $B^1(\mathbb{R},\mathbb{R})$ be the set of all locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $$\sup_{t\in \mathbb{R}} \int_t^{t+1}|f(x)|dx<\infty.$$ Consider the map ...
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1answer
42 views

Gelfand triple: what happens if we don't identify the pivot Hilbert space with its dual?

People usually say $V \subset H = H^* \subset V^*$ is a Gelfand triple if $V$ is continuously and densely embedded in $H$ and $H$ is identified with its dual. Sometimes they do not mentioned that ...
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55 views

$B$-tight frame (Tao and Kadison Singer)

I am just trying to understand this text from Tao: $\renewcommand{\C}{\mathbb C}$ Assume you have $w_1,\ldots,w_m \in \C^d$ and $B > 0$ with $$ \sum_{i=1}^m |\langle w_i, u \rangle |^2 = B $$ for ...
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99 views

A question about convex open set in a topological vector space.

Supose $E$ is a topological vector space(may not be Hausdorff). $U\subset E$ is an open set such that $U+U=2U$. How to show $U$ is convex? I can see if $E$ is $T_1$,then $E$ should be Hausdorff. ...
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1answer
53 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
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1answer
92 views

Banach fixed point theorem (application)

Let $X$ and $Y$ be Banach Spaces, and $T: X \to Y$, where T is continuous, linear and bijective, and let $S: X \to Y$ (where $S$ is continuous and linear) with $|S|\cdot|T^{-1}|< 1$. Show that ...
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34 views

exact angle between the functions $p(x)=3x-4$ and $q(x)=9x-5$ over $0\leq x\leq 1$

I was attempting a question, which gave the formula for the angle theta between two functions $f(x)$ and $g(x)$ over $a\leq x\leq b$ (note the question defined the meaning of norm and inner product as ...
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1answer
28 views

Characterization of multipliers on the Dirichlet space of holomorphic functions

Can someone tell me whether there is a characterization of boundedness of multiplication operators on the Dirichlet space of the unit disc? These are holomorphic functions $f$ on the unit disk $D$ ...
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1answer
79 views

Is there a complete orthornomal basis of a Hilbert space which takes positive values on a discrete set?

Is there a complete orthonormal basis $\{f_n\}$ (of continuous functions) of the Hilbert space of square integrable functions on $[0,\,\infty)$ for which there exists a countable set $S\subset ...
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1answer
51 views

Integrated Brownian motion: independent stationary increments?

Let $B_t$, $t\in [0,T]$ be a $d$-dimensional standard Brownian motion. Let $\sigma:[0,T] \rightarrow \mathbb R^{d\times d}$ be a deterministic function such that $$\sigma(u) = diag( \sigma_1(u), \dots ...
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54 views

Operator for scaling a function?

Let $\mathbb{F}$ denote the set of functions of the form $f: \mathbb{R} \to \mathbb{R}$. I am interested to know whether there exists a well-known linear map $T_\alpha: \mathbb{F} \to \mathbb{F}$ ...
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110 views

Selfadjointness of the Dirac operator on the infinite-dimensional Hilbert space

I am a physicist, so my background in functional analysis is limited only to basics. However, I would like to prove that the free Dirac operator is selfadjoint (or Hermitian, or neither). The free ...
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1answer
125 views

What's Helly's theorem in the proof of the Goldstine–Weston density theorem

I have a problem in understanding a proof of Goldstine–Weston density theorem. The only thing I don't know in the proof is the part of Helly's theorem to be related. The Goldstine–Weston density ...
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1answer
70 views

Failing to reproduce specific Functional derivative

I'm failing to reproduce an (indirect) result in a paper, namely $${δF[g]\overδg(x,y,z)}={r^4\over\ell^5} $$ where $F[g]=\iiint \frac{2dxdydz}{\ell g(x,y,z)} $ and $g(x,y,z)=-{\ell^2 \over r^2} $. ...
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1answer
64 views

Continuity of a composition map between Holder spaces

Let $\varphi\in C^{\infty}_0(\mathbb{R})$, $0<\alpha<1$, $\Omega\subset\mathbb{R}^d$ be a bounded domain. Is it true that the map $\Phi:C^{0,\alpha}(\bar{\Omega})\to ...
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1answer
300 views

norm of inverse of a bounded operator [duplicate]

Are there any conditions in which norm of inverse of a bounded operator T is equal to reciprocal of norm of the operator T.
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1answer
87 views

Boundary conditions for the time-independent Schrödinger equation on the sphere

if you have a free Schrödinger operator on a sphere $-\Delta \psi(\theta,\phi) = E\psi(\theta,\phi),$ then the substitution $\psi(\theta,\phi) = f(\theta)e^{i n \phi}$ leaves you with the ...
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1answer
55 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
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1answer
57 views

Approximate unit of an involutive Banach algebra

I know that every C*-algebra has an approximate unit. I have two questions: why we cannot show that every involutive Banach algebra has an approximate unit? I need an example of an involutive Banach ...
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1answer
49 views

Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
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1answer
79 views

The spectrum of $L:=-\Delta+V(x)$ on complex $L^2(\mathbb{R}^N)$ and real $L^2(\mathbb{R}^N)$

In general, when one talks about the spectrum of an self-adjoint operator, it is naturally considered in a complex Hilbert space (say $L^2(\mathbb{R}^N,\mathbb{C})$). Moreover, the spectral ...
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1answer
53 views

Positive elements in a Banach algebra

Let $A$ be a unital Banach algebra. If $a$ is an element of $A$ with $||1-a||_{sp}<1$, then there exists $b\in A$ such that $b^2=a$. Furthermore, if $A$ is an involutive Banach algebra and if $a$ ...
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56 views

On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
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1answer
62 views

Non-reflexive Banach space implies the dual is non-reflexive.

So here is my question: If $X$ is a Banach space that is not reflexive, then $X^*$ is also not reflexive. I have looked at the answer to the following: A question about Banach reflexive space The ...
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3answers
43 views

New metric spaces from given one

Let (X,d) be a metric space, $f : [0,\infty) \rightarrow [0,\infty)$ continuous differentiable, strict monotone increasingly with $f(0)=0$ and a monotone decreasingly derivative. Prove that $f \circ ...
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43 views

How to calculate Hill's discriminant?

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
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1answer
31 views

Convergence in $L^1 \cap L^2$

I am very confused about the following: Assume we have a sequence of functions $f_n \in$$L^1 \cap L^2 (\mathbb{R}^n)$. Then is it true that if this sequence is Cauchy both in $L^1$ and $L^2$, two ...
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1answer
32 views

Are $l_{p} \cap k$ and $l_{p} \cap k_{0}$ complete in $||$ $||_{\infty}$? Are they complete in $l_{p}$ norms?

Let the space $k$ be all convergent sequences of real numbers. Let the space $k_{0}$ be the space of all sequences which converge to zero with $l_{\infty}$ norm. Are $l_{p} \cap k$ and $l_{p} \cap ...
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35 views

distance between two space

‎Given N- dimensional subspace ‎$W‎_{1}‎$ ‎and ‎‎$W‎_{2}‎‎$ ‎of a‎ ‎Hilbert ‎space‎, define the ‎$‎N$-tuple $(‎σ_{1},σ_{2},...,σ_{N}) $‎‎‎ as follows: ‎$$\sigma_{1}=\max \lbrace\langle ...
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2answers
84 views

Compact projection on Banach space has finite rank

Let $E$ be a Banach space. Show that every compact projection has finite rank. I have no idea where to start.
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1answer
52 views

Nonequivalent norms on infinite-dimensional linear space

I've just proven that for every infinite-dimensional space with a norm $(V, ||~||)$ we can find a discontinuous linear functional $ \phi $. But next I'm trying to prove the following: The norm $ ...
3
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1answer
34 views

Family of sequences in a Hilbert space with certain property

Suppose $\mathcal{F}$ is a family of sequences on the unit sphere of $l_2$ with the following property: For any sequence $\varepsilon_n\downarrow 0$ but which is not eventually identically $0$, there ...
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2answers
28 views

Functions in $L^p(\mathbb{R}^n)$, are tempered distributions.

How to prove that functions in $L^p(\mathbb{R}^n),1 \leq p \leq \infty$, are tempered distributions.
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3answers
250 views

Hermitian and self-adjoint operators on infinite-dimensional Hilbert spaces

I am a physicist and I am trying to get a grasp on the following terms from functional analysis: As I understand, an operator is Hermitian if it is symmetric and bounded (domains of A and A* don't ...