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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Density of a subset of a Hilbert space

I've been trying with a colleague but we could not come to a solution. The problem is as follows: Let $M$ be a subset of a Hilbert space $H$, and let $v,w\in H$. Suppose that $\langle ...
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show that if $y$ is orthogonal to $x_n$ and $x_n$ converges to $x$ then $x$ is orthogonal to $y$

help me. someone who can help me? spaces is inner product. It is section 3.2, issue 4 introduction to functional analysis book author Kreyszig
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33 views

Inverse Function Theorem for Banach Spaces

In the middle of a proof of the Inverse Function Theorem (namely, the proof of Baby Rudin), we use the fact that if $A$ is invertible and: $$ ||B-A||~||A^{-1}|| <1$$ then $B$ is invertible. The ...
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1answer
38 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
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2answers
30 views

Continuity and differentiability of $f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!(x+n)}$

Given the series: $$\sum_{n \ge 0} \frac{(-1)^n}{n!(x+n)}$$ Let $f_n(x)$ denote its general term. Let $f(x)$ denote its sum (when exists). The question asks to: $i)$ Find the domain $\mathbb D$ on ...
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14 views

Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
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13 views

Hilbert space and Orthogonal complement

Let $M$be a linear subspace of Hilbert space $H$. Show that $$cl M = H \Leftrightarrow M^{\perp} = \left\{ 0 \right\}$$
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2answers
25 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
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33 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
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0answers
22 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
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13 views

convex function with Hessian measure $D^2 f \leqslant \lambda$ $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
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1answer
35 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
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0answers
42 views

Is the linear operators must be invertible to from a category?

I am trying to understand the concept of category in mathematics. For example the following link talks about category $Lin$ which is an Abelian category. ...
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1answer
31 views

$L^1([0, 1]) \subset C([0, 1])^*$

Basically my question is: how can I prove that $L^1([0, 1]) \subset C([0, 1])^*$, where $C([0, 1])$ represents all continuous functions on $[0, 1]$, and the superscript $^*$ means the dual space. ...
2
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0answers
19 views

An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
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1answer
28 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
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64 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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42 views

Example of projection sequence on Hilbert space with strong limit P

Let $P_n$ be strongly convergent with limit $P$, where $P_n$'s are projections on a Hilbert space $H$.Suppose that $P_n(H)$ is infinite dimensional. Show by example that P(H)$ may be finite ...
3
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1answer
46 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
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1answer
32 views

Invertible , bounded linear operator on a Hilbert space

Suppose we have an invertible, bounded linear operator $K$ on a Hilbert space $H$. Is there a constant $c \in \mathbb{R}_+$ such that $$ ||Ku|| \geq c||u|| $$ for all $u \in H$ ?
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26 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
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1answer
18 views

Prove that Euler's equation can be written in a specific form

According to my notes, the following theorem holds: If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ ...
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Prob. 8, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS [on hold]

if $\|x+ \alpha y\|\ge \|x\|$ for all $\alpha$, then show that $x$ and $y$ are perpendicular. If you could explain this I shall be thankful....
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36 views

Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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16 views

Why $q(]-1,1[)=B(q,0,1)$.

Let $(X,\mathcal T)$ a topologique space defined by $\Gamma$ a set of semi norm. Prove that if $q:X\to \mathbb R$ is a continuous semi norm, then $$\exists p\in\Gamma_f,\exists \alpha>0:q\leq ...
2
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1answer
20 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
1
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1answer
27 views

Problem with the Definition of contractible set

I have this definition of contractible set: we say that $A\subset X$ is contractible in $X$ if there exists a continuous function $\eta:[0,1]\times A\rightarrow X$ such that $\eta(0,x)=x, \forall ...
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The relation between the (algebraic) dimensions of a normed linear space and its dual.

What is the relation between the (algebraic) dimensions of a normed linear space and its dual, for example can we say $\dim X \leq \dim X^*$, for a normed linear space $X$?
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12 views

Unitary operator with absolutely continuous spectrum

I have a unitary operator $U_0$ acting on $H := L^2(\mathbb{T}=[-\pi,\pi]; \mathbb{C})$, denote its spectral family by $\{ E_0(\cdot) \}$. Moreover, the spectrum of $U_0$ is purely absolutely ...
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36 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
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0answers
29 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
0
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1answer
24 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
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0answers
19 views

Is it possible to modify norm of Sobolev space suitable for ill-posed problems.

I have trying to pose my problem mathematically for quite some time now. I am not sure even if I am close to defining properly. Would anyone please help: I am interested to study ill-posed problem of ...
1
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1answer
38 views

Countable set in a Banach space which spans densely?

Let $\mathcal{C}(\mathbf{T})$ be the algebra of continuous complex functions on the unit circle $\mathbf{T}$. Consider the following two statements: The $*$-subalgebra generated by $1$ and $z$ spans ...
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functional analysis. Riesz system. [on hold]

I have the following problem: "a) Let $H$ --- Hilbert space, $T : H \to H $ --- unitary operator. For the vector $x_0 \in H$, we assume that $x_j := T^j x_0, j = 0,-1,+ 1,-2,+2 ... $ suppose that ...
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28 views

Is there any relation Trace and Boundary?

I understand the trace is sum of diagonal elements of a matrix. Further the boundary I always perceive as a 'end points' of bounded domain. However on the link below: ...
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2answers
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Does $a_n \in H^1(\mathbb R^n)$ and $b_n \rightharpoonup 0$ in $H^1(\mathbb R^n)$ imply $\langle a_n, b_n \rangle \to 0$?

I have a question mainly in functional analysis. Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows: Does $a_n \in H^1(\mathbb R^n)$, $|a_n| ...
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0answers
16 views

unique inner product on a tensor product of Hilbert $C^*$ modules and Hilbert spaces.

For a $C^*-$ algebra $A$ and a Hilbert space $H$ and a Hilbert $A-$module E; how can we show that there is a unique $A-$ valued inner product on $H \otimes E$ as $< h_1 \otimes x_1 , h_2 \otimes ...
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1answer
62 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
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1answer
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State of a $ C^{*} $-algebra.

Let $ (\pi,\mathcal{H}) $ be a non-degenerate $ * $-representation of a $ C^{*} $-algebra $ A $, and let $ h \in \mathcal{H} $ with $ \| h \| = 1 $. Define $ f_{h}: A \to \Bbb{C} $ by $ {f_{h}}(a) ...
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1answer
31 views

What does this notation mean? Functional Analysis

I am studying analysis at the moment and came across this notation and I would like to know what it really means: $$C_{c}^{\infty}(\Omega)$$ My understanding so far is that,this is the space of ...
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1answer
43 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
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15 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
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Application of Polish Space and Lebesgue measurable.

I understand Polish space is useful for non-Lebesgue measurable set but is it also applicable for Lebesgue measurable set?
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1answer
25 views

Every nontrivial linear functional is open

Let $X$ be a normed linear space and let $f:X\to \mathbb K$ be a nontrivial linear functional. I want to prove that $f$ is open. I tried as follows: Let $E$ be an open set in $X$ and let $y\in f(E)$. ...
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38 views

Derivation of perturbation series

I'm a little bit confused about the derivation of the perturbation series. I know from my quantum mechanics course that for a perturbed operator, eigenvalue is a series that is depend on the ...
3
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1answer
28 views

Reducing a double ultrapower to a single ultrapower

I hate having to ask this question, as I know for a fact I have seen the answer before but cannot seem to find it. So I'm breaking down and asking for a reference. Given a structure, let's say a ...
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14 views

Relation between Ill-posed problem and eigenvectors?

This question is related to the question below: Is there a relation between Ill-posed problems and Eigenvectors. In the answer of the above question, it was shown that the ill-posed problem can be ...
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21 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G(\int k(x,y)f(y)dy)$ $(*)$ where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...
4
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1answer
24 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...