Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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4
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2answers
17 views

Given a normed vector space $X$, can you always define a bounded linear functional $f$ which is bounded above and below by the norm?

Let $X$ be a (possibly infinite-dimensional) normed vector space over $\mathbb{R}$, whose norm is denoted $|| \cdot ||$. Given two non-negative scalars, say $M, N \in \mathbb{R}$ with $M < N$, can ...
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7 views

Reference for Closure of Nilpotent limit (Operator Theory )

I'm looking for a proof of this theorem. The [13] in the picture is "Approximation of Hilbert Space Operators" Volume 1 by Herrero. But I couldn't find any available textbooks online at the ...
4
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1answer
65 views

Prove $\tan \left( \frac\pi2 \frac{(1+x)^2}{3+x^2}\right) \tan \left( \frac\pi2 \frac{(1-x)^2}{3+x^2}\right)\le\frac13 $

How to determine the range of the function given by $$f(x)=\tan \left( \frac\pi2 \frac{(1+x)^2}{3+x^2}\right) \tan \left( \frac\pi2 \frac{(1-x)^2}{3+x^2}\right) $$ It is directly observable that the ...
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1answer
9 views

Relations between supremum, infimum the basis of Banach space and its associated sequence of coefficient functionals

Let $E$ be a Banach space with a basis $\{x_n\}$ and let $\{f_n\}$ be the a.s.c.f. Then The coefficient functionals $\{f_n\}$ associated to the basis $\{x_n\}$ are continuous linear functionals on $E,$...
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0answers
22 views

Fractional Power of a linear operator?

This question asks about fractional powers of matrices: Fractional power of matrix Is there a generalization to (e.g. continuous or bounded) linear operators (which I like to think of as infinite-...
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0answers
21 views

Meaning of degenerate PDE

I am trying to understand the concept of degenerate PDE. The definitions I found depend on the type of equation but let say for elliptic (see here). Let's take an example: $$x\partial_x^4u + \...
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0answers
20 views

Looking for references: in PDEs

This is not a technical mathematical question. I came across some PDEs with no references nor their names. $$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \tag{Eq1}$$ The above equation can ...
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0answers
8 views

Question about Proof of Wold's Decomposition Theorem (Operator Theory ) on Wikipedia

(Wold-decomposition theorem 1954.) Every isometry is a direct sum of unitary and unilateral shifts. Sketch of Proof: https://en.wikipedia.org/wiki/Wold%27s_decomposition#A_sequence_of_isometries In ...
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2answers
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Continuously applying an isometry operator on to a Hilbert space

While looking at the proof of https://en.wikipedia.org/wiki/Wold%27s_decomposition#A_sequence_of_isometries, I encounted that if $V \in B(H)$ is an isometry on a Hilbert space $H$. Then we have $V^n H ...
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1answer
32 views

Defining a mapping from a set to a set

How to define a map that takes a finite binary string of length $L$ to a real number in $[a,b]$ where length $L$ is the number of bits constituting the string $x=0101$ is a binary string of length $L=...
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1answer
67 views

Why doesn't the lack of pointwise convergence of $f_n(x)=x^n$ contradict the fact that $C[0,1]$ is Banach?

On this answer, the function $f_n(x)=x^n$ in the interval $[0,1]$ is given as a pathologic example with pointwise convergence. Can I say that this Cauchy sequence does not (pointwise) converge because ...
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1answer
29 views

Reference for Wold-Decomposition Theorem (Operator Theory)

I'm looking for full proves for the following theorems: (Wold-decomposition theorem 1954.) Every isometry is a direct sum of unitary and unilateral shifts. Every isometry is unitary equivalent to a ...
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0answers
16 views

Positive Operator on $C(X)$ with norm 1

Let $X$ be a compact set, $C(X)$ the space of continuous function with values in $\mathbb{R}$ with the usual norm (uniform convergence) and an operator $T: C(X) \to C(X)$ with these conditions 1- ...
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1answer
29 views

Understand rank-one operators in Hilbert space

Let $\{e_n \}$ be an orthonormal basis on a Hilbert space $H$. When we say that $T$ is a rank-one operator, it means that the range of $T$ has dimension $1$. Then for example, would $Te_n=e_n$ be of ...
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21 views

How can I approximate $L ^ 1(0,1)$ functions with continuous functions?

Is It possible to approximate functions on $L^1(0,1)$ whit continuous function?
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0answers
20 views

reference for Poincaré constant of $W_0^{1,1}(B)$

I'm trying to figure the optimal Poincaré constant of the space $W_0^{1,1}(B)$, Sobolov space on the unit ball in $\mathbb{R}^n$. Most paper I found is either about $W_0^{1,2}$ or $W^{1,2}$ with $0$ ...
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1answer
18 views

Show two topologies coincide on the unit ball.

Consider the following lemma from "Lectures on von Neumann algebras": I understand the proof of $(i)$ and $(ii)$. However, the proof says that $(iii)$ and $(iv)$ follow immediately from $(i)...
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Find the all spectrum [closed]

Find the all spectrum of the next operator \begin{align} A:& L_{2}\longrightarrow L_{2} \\ & f \longrightarrow A(f)=\alpha f \end{align} where \begin{equation*} \alpha(t)=\left\{\begin{...
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0answers
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Clarification about extensions of Ornstein-Uhlenbeck operator

I am reading stuffs regarding the Ornstein-Uhlenbeck operator and its various extensions to $L^p(\gamma)$, with $p \in (1,+\infty)$ and with $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$. ...
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3answers
52 views

How to prove $(P-\lambda)^{-1} = \frac{1}{\lambda(1 - \lambda)}P - \frac{1}{\lambda}I$?

I have a question regarding Functional Analysis: Assume that $X$ is a Banach space and that $P \in B(X)$ is a projection with $ran P \neq {[0]}$. Then I have easily shown that $\lambda = 0$ and $\...
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1answer
37 views

Is uniform convergence equivalent to Cauchy sequences for the space of continuous functions on an interval?

Having received a perfect answer to this question, I am left trying to revisit what I thought I grasped at an intuitive level. The present question asks for clarification on the comment: The example ...
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1answer
26 views

Why is this functional norm-continuous?

Consider the following fragment from the book "Lectures on von Neumann algebras": Why is the marked line true? I'm probably missing some basic functional analytic fact here.
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2answers
32 views

Question about initial topology and dual vector space

Consider the following fragment from the book "Lectures on von Neumann algebras". Why is the line $\varphi$ is $\sigma(\mathcal{E}, \mathcal{F})$-continuous $\implies$ there exist $\psi_1, ...
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Comparing decompositions for two non-orthogonal bases in Hilbert Spaces

Let $H$ be a real separable Hilbert space and $B^1=\{w^1_j\}_{j \in J}$ and $B^2=\{w^2_j\}_{j \in J}$ two basis sets. Consider the linear maps \begin{align} P^i_j: H& \longrightarrow \mathbb{R}\\ ...
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1answer
32 views

Is there a complex Geometric Hahn Banach theorem for imaginary part?

I know the complex version of the geometric Hahn Banach theorem, e.g. in here https://math.stackexchange.com/a/2497406/789584 Let $X$ be a (complex) locally convex space, and $A,B\subset X$ disjoint, ...
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0answers
17 views

Semilinear Abstract Differential Equations, Mild solutions.

Reading Chapter 6 of Pazy book, I realize that the nonlinear perturbation $f(t, u)$ is everywhere assumed to be continuous with respect to $t$. In particular, I am talking about mild solutions to an ...
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0answers
35 views

Elliptic problem with gradient term: $-\Delta u + v\cdot \nabla u = 0$ with $v \in \dot H^1$ [closed]

Let us consider the problem $$ -\Delta u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$ where $v:\mathbb R^n \to \mathbb R^n$ satisfies $$\int_{\mathbb R^n} |\nabla v|^2 dx < +\infty$$ ...
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0answers
33 views

On the Spectrum of $-\Delta+V$ and the Image of $V$

Both the fundamental examples of Schrodinger Operators (Harmonic Oscillator, Hydrogen Atom ones) and the physical intuition suggests that the discrete spectrum of $-\Delta+V$ is always a subset of the ...
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1answer
27 views

A best approximation $v ∈ V$ of $x$ exists, then show that $v$ is unique.

I have a question regarding functional analysis: Assume that $X$ is a strictly convex normed linear space and that $V ⊂ X$ is a proper linear subspace. In addition, assume that for some element $x ∈ X$...
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0answers
19 views

Example of a rank-one operator to satisfy conditions about the spectrum

Let $U$ be a bilateral shift operator such that $Ue_n= u_ne_{n+1}$ for $n\in \mathbb{Z}$. I am looking for an example of a rank-one operator $T \in \mathcal{B}(\ell^2(\mathbb{Z}))$ so that $\sigma (U) ...
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1answer
51 views

How do I show that the map $u \to \int u^2$ is weakly continuous [closed]

I am new-ish to the material. How do I show that the map $u \to \int u^2$ is weakly continuous in the space is $H^1_0(\Omega)$ using the fact that $H^1_{0}(\Omega)$ is compactly embedded in $L^2(\...
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2answers
43 views

What are the elements of metric space?Is it always an interval are it can be set of integers etc.

What are the elements of metric space? Is it always an interval or it can be set of integers, etc.?
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1answer
25 views

A bilateral weighted shift $T$ with $\sigma(T) =$ unit circle

I am looking for an example of a bilateral weighted shift $T\in B(\ell^2(\mathbb{Z})) $ with $\sigma(T) = \mathbb{T}$, which is the unit circle. NOTE: what I'm looking for a bilateral weighted shift ...
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1answer
22 views

Clarification regarding the space of continuous functions and its completeness (Banach)

A Banach space is a normed vector space with completeness, i.e. all Cauchy series converge to an element within the vector space. In the case of continuous functions on an interval in the real line, $...
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0answers
16 views

Calculating basis constant of coefficient functionals

Let's assume we have a Banach space $X$ over field $\mathbb{K}$ with a Schauder basis $(x_n)_{n=1}^\infty$. We define the following sequence of functionals $f_n: X \rightarrow \mathbb{K}$ by $$\langle ...
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1answer
18 views

orthogonal complement to range of a bounded operator closed iff bounded

Let $H,K$ be Hilbert spaces. Suppose $T\in B(H,K)$. Consider the restriction map of $T$, $T': \mathcal{N}^{\perp}(T) \rightarrow \mathcal{R}(T) $ . Show that $\mathcal{R}(T)$ is closed if and only if ...
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0answers
34 views

Normed Space $L^2$ and Vector Space $l^2$

I am really confused about $L^2$-Space and $l^2$-Space. Can anybody explain what is the difference between the $L^p$ and $l^p$ space? Also in $L^p$-Spaces, we see that $L^2 \subset L^1$ (for $L^2[(0,...
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0answers
38 views

Knowing the number of negative eigenvalues of a $ A $ operator, is it possible to know of the $ -A $?

Let $H=(H, (\cdot, \cdot)_H)$ be a Hilbert space and $A: D(A) \subset H \longrightarrow H$ be a linear, densely defined and self-adjoint operator. Suppose that the spectrum $\sigma(A)$ of $A$ is ...
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0answers
33 views

Is the $L^2$ norm bounded by the $H^{1/2}$ norm?

I know that when s is an integer then $\|u\|_{L^2(\Omega)}\leq \|u\|_{H^s(\Omega)}$. However, the definition for the fractional Sobolev spaces is $u \in L^2(\Omega)$ such that $\frac{\|u(x)-u(y)\|}{\|...
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2answers
18 views

Scaling a compact gives a compact in a locally compact vector space

While wondering about why locally compact vector space are important, I was told that in a locally compact vector space if one scales a compact set by some number, than we still have a compact set, ...
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1answer
23 views

Solve $u_t+cu_x=0$ with $c>0$ when $x,t >0$ and contions $u(x,0)=h(x)$ with $x>0$ and $u(0,t)=g(t)$ with $t>0$.

$u_t+cu_x=0$ show that $u(x,t)=f(ct-x)$. With conditions, it shows that $$u(x,0)=f(-x)=h(x),$$ $$u(0,t)=f(ct)=g(t)$$ I dont know how to find what $f(ct-x)$ is and what the purpose of condition: $c&...
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0answers
17 views

Norm of infinite (almost) stochastic matrix

Let $A$ be an infinite matrix $$A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots \\ a_{3,1} & a_{3,2} & a_{3,3} & \...
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1answer
34 views

Lemma 7.8, Applied Analysis (Hunter)

Here's an image of the lemma and the relevant definition preceding it: I'm trying to understand the approach to the proof. It's not clear to me why it's sufficient to prove that $$ ||S_M - S_N||_\...
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0answers
29 views

Help me please to resolve this property in $L^!$ space!!

This exercise is similar to other proposed oh the forum, but it's different because the function $\psi$ is in $L^1$ space and not in space of continuous functions. $v(x) = \begin{cases} \xi_1 &...
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1answer
25 views

Give an example of a set that is closed under a norm and not closed under another

Q: Give an example of a vector space $E$ and a subset $F$ of $E$ such that $F$ is closed under a norm topology and not closed under another norm topology. I tried to find an example but i was not able ...
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1answer
19 views

Derivative of the expected value of the truncated normal distribution wrt upper limit

I am looking for the derivative of the expected value of the truncated normal distribution with respect to one of the upper limits. It looks like this: $$f=\frac{1}{\sqrt{\vert T \vert} (2\pi)^{3/2}} \...
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1answer
28 views

Does the weak closure of a set S always contain the convex hull of S (infinite dimension)?

Let $X$ be an infinite dimensional normed vector space. I learned that the weak closure of $S^1=\{x\in X:\|x\|=1\}$ is the ball $\{x\in X:\|x\|\le 1\}$. I was wondering if the weak closure always ...
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2answers
26 views

Bidual of Banach Space

I am trying to prove that given a reflective Banach space $X$, i.e $X^{**}=X$, and a complet subspace $E \subseteq X$ then $E$ is reflective to. I had som problem proving it so i looked at a solution ...
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1answer
54 views

Show that a certain net converges in a von Neumann algebra.

Consider the abstract von Neumann algebra $M = \ell^\infty\text{-}\bigoplus_{i \in I} B(H_i)$. Moreover, we assume $\dim H_i< \infty$ for all $i \in I$. Let $x_i$ be the identity on $B(H_i)$ and ...
2
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1answer
18 views

Does the limit of gradient flow from a given initial value depend on the choice of inner product?

I will start with an example to motivate my question, and then ask it more generally. Example. Let us consider the Hilbert spaces $H^1 \subset L^2$, and a Frechet-differentiable functional $E:H^1\to\...

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