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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Question about proof of Tychonoff-Alaoglu

I'd like to check that I understand the proof in full detail. Can you tell me if the following is correct? Thanks for your help. Claim: The closed unit ball $B_{\|\cdot\|_{op}}(0,1)$ in $X^\ast$ is ...
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146 views

Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
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399 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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115 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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435 views

Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?

From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$, the set of real-valued continuous functions on X, is ...
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268 views

Weakest topology with respect to which ALL linear functionals are continuous

One often considers a Banach space $X$ under the "weak topology", ie. the weakest topology such that all bounded linear functionals are continuous. This leads me to wonder about the weakest topology ...
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73 views

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many ...
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35 views

Is the Banach space of continuous functions on a compact space with a coutable base separable?

Let $X$ be a compact Hausdorff space with a countable base. Let $C(X)$ be the Banach space of complex valued continuous functions on $X$. Is $C(X)$ separable, i.e. does it have a countable dense ...
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45 views

Composition of function with linear functional is Lipschitz implies function itself is Lipschitz

This is taken from Conway's A course in functional Analysis (p. 98, Exercise 9): If $(S,d)$ is a metric space and $X$ is a normed space, show that if $f:S\rightarrow X$ is a function such that for ...
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32 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
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93 views

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
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45 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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33 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
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55 views

Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
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54 views

Modifying a smooth function with respect to conditions on its partial derivates

Let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets of $\mathbb{R}^n$, $K_i\subseteq U_i$ compact subsets, $\epsilon_i>0$ positive real numbers and a nonnegative natural number ...
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45 views

What does a well-posed problem imply?

A well-posed problem in the sense of Hadamard states that: A solution exists The solution is unique The solution's behavior changes continuously with the initial conditions. Now in order to prove ...
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50 views

Why $\|f-g\|=0$ if and only if $f=g$?

I'm learning Fourier Transformation lately, and in the Course Reader page 23, it defines $\|f\|=\left(\int_0^1 \left|f(t)\right|^2 dt\right)^{1/2}$. And then $\|f-g\|=0$ if and only if $f=g$. My ...
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78 views

What is the relationship beteen functional analysis and topology

Could someone explain in layman's way using examples how is topology related to functional analysis? (edit) After taking a UG course in Point-set topology i felt to have a taste of functional ...
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35 views

Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra

I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra. Thinking about it for 10 minutes, I didn't ...
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52 views

is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
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64 views

why $ \nabla v_n \to \nabla v \ \ (a.e.)$ and $ v_n \to v $

Can someone see the 10th line of page 9 in this article and give a hint that why $$ \nabla v_n \to \nabla v \ \ (a.e.)$$ and $$ v_n \to v $$ and how with theorem 2.1 we could conclude there exists $ ...
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46 views

The weak-star topology is Completely Hausdorff (in particular Hausdorff).

Let $X$ be a normed space, $X^*$ its dual space, $(X^*, w^*)$ is completely Hausdorff. Proof: Let $f, g \in X^*$, $f\neq g$ then $\exists x\in X$ such that $f(x)\neq g(x)$ i.e. $\hat{x}(f)\neq ...
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89 views

Relationship between functional analysis and differential geometry

I am taking courses on functional analysis (through Coursera.com) and differential geometry (textbook author : O'neil) on my university. I made the following table on my own. Are the similar ...
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226 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
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168 views

On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
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80 views

Is exponential function in a C*-algebra injective on self-adjoint elements?

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?
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45 views

Existence of functional implies non-negative supremum.

$\ell_\infty(T) = \{f:T\to\mathbb{R} : \sup_{t \in T}|f(t)| < \infty\}$. Let $G$ be a family of transformation of set $T$ and $V$ linear subspace of $\ell_\infty(T)$ such that: $1_T \in V$, ...
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63 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
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69 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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87 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
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58 views

Probability measure on the set of probability measures

Let $X$ be a metric space, $\mathcal{P}(X)$ be the the set of all Borel probability measures on $X$. I endow $\mathcal{P}(X)$ with the weak* topology. Question 1: Do I need to impose some further ...
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68 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
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33 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
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73 views

Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
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59 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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72 views

Limit of distribution

Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that $$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g ...
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111 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
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votes
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74 views

Proof of a theorem about Baire categories

Problem: prove that the set of $C([0, 1])$ functions whose derivative is defined at every point (and it is either finite or infinite) is of the first Baire category. I have no idea how to approach ...
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36 views

Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
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68 views

Does this characterize the operator norm of the inverse?

Let $A$ be an invertible operator (bounded with bounded inverse). Then $$\frac{1}{\|A^{-1}\|} = \inf\left\{\frac{\|Av\|}{\|v\|} : v \neq 0\right\}$$ I believe I have a proof as follows, but I just ...
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128 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
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55 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
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255 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
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155 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
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111 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
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307 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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245 views

On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...
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72 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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61 views

Compatible maps and Sub- compatible maps

Definition Let $M$ be a subset of a metric space $X$. The set $M$ is called $q$- starshaped if $ kx+(1-k)q \in M$ for all $x\in M$ and $k\in[0,1]$, where $q$ is an element of $M$. Definition Let $M$ ...
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48 views

Are pseudo(micro)-local operators pseudodifferential?

$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the ...