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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51 views

Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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39 views

Finding norm of functional coefficents in $C[a,b]$

Let $\{x_n\}$ denote Schauder’s basis for $C[a, b]$ and let $\{h_n\}$ be the associated sequence of coefficient functionals. Compute $||h_n||$. In young's book, a schauder basis for $C[a,b]$ ...
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53 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
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94 views

p-direct sum and dual spaces

I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(. Let E and F be normed spaces. For $p \in [1,\infty]$, define the ...
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17 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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49 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
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86 views

Parabolic holder norms

Let $Q=\Omega\times[0,T]$ be a cylinder with $\Omega$ bounded open set in $\mathbb{R}^N$. N.V.Krylov in "lectures on elliptic and parabolic equations in holder spaces" defines the parabolic holder ...
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105 views

Definition and Intuition of a Weakly Dense Set

What does it mean to say: set A is "weakly dense" in a set B? The definition of a "dense set" is rather intuitive: the classic example of Q (rationals) being dense in R (reals) is very clear. How ...
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34 views

Existance of the integral in the domain of generator of the strongly continuous semigroup

Let $\{s(t)\}_{t\geq 0}$ is a $C_0$ semigroup of bounded operator on the Banach space $X$ and $A:D(A)\subset X\rightarrow X$ be the infinitesimal generators of the semigroup $\{s(t)\}_{t\geq 0}$. ...
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64 views

Operator on the space of square summable sequences

We define an operator $T:\mathcal{l}^2(\mathbb{Z})\rightarrow\mathcal{l}^2(\mathbb{Z})$ where $\mathcal{l}^2(\mathbb{Z})$ is the Hilbert space of square summable functions, such that for ...
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29 views

functional calculus on a set of normal elements is continuous

Let $K$ be a compact subset of $\Bbb C$. Let $A_K$ denote the set of all normal elements $x$ with $\sigma_A(x)\subset K$. If $f$ is a continuous function on $K$, then the functional calculus :$x\in ...
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41 views

Linear operator with respect to matrices? Is it natural to ask?

Let $T:E\rightarrow \mathbb{R}^{n\times m}$ be a linear operator where $E$ is some Banach (functional) space. Then for every $\lambda \in \mathbb{R}$ and $x\in E$ one has $T(\lambda x)=\lambda T(x)$. ...
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43 views

Find a operator from $L^2$ to $L^2$ satisfying a given condition

My question is as follows: For $k\in\mathbb{N}$, $k\ge 2$, I need to find an operator $A: L^2(\mathbb{R}\to L^2(\mathbb{R})$ such that for all $u\in L^2(\mathbb{R})$ we have $[(A^* ...
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56 views

Proving that the set is closed.

We use the sequential definition to prove a set is closed. So no continuity or closure or anything related to the topology of the set is allowed. Show $A = \{ x \in \ell^2: |x_n| \leq 1/n \}$ is ...
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53 views

Can the Hamburger moment problem be solved for probability measures?

The hamburger moment problem states that given any real sequence $\{a_n\}$, there exists a positive Borel measure $\mu$ such that $$ a_n =\int_{\mathbb{R}} x^{n}\,d\mu. $$ In other words, the ...
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115 views

What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
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98 views

Minimum distance from a subspace in a Banach space

$X,Y,Z$ are disjoint sets of vectors in a Banach space M. $S_X,S_Y,S_Z$ denotes the subspace formed by the vectors in $X, Y$ and $Z$ respectively. $S_{YZ}$ and $S_{XZ}$ denote the subspaces formed by ...
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29 views

product of one dimensional basis functions spanning two dimensional space

Lets assume I have a set of basis functions $h_1(x),h_2(x), ...$ spanning the whole hilbert space of one dimensional square integrable functions. Now I want a basis set that spans the the whole ...
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31 views

$\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$ dense in $C_0(\mathbb{R}_+)$.

Let $\mathbb{R}_+ := [0,\infty )$ and let $w \in C_0(\mathbb{R}_+)$ be any function with $w(x) \neq 0$ for all $x \geq 0$. Why is $\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$, where $e_t(x) := ...
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37 views

Prove that perfect set is not countable

A subset $S$ of a metric space $(X, d)$ is called perfect if it is closed and every point of S is an accumulation point of S. Prove that if S is a nonempty perfect subset of a complete metric space, ...
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78 views

Prove that intersection of nested closed sphere is not empty without $r_{n}\to 0$.

Given a Banach space E, let $B_{p_n}(z_{n})= \left\{x\in E: ||x-z_{n}||\leq p_{n}\right\}$, $z_{n}\in E$, $p_n\geq 0$, $n=1,2,...$ be a sequence of closed balls in $E$ such that ...
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51 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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20 views

The absolutely continuous representative of $t \mapsto (u(t), v(t))_{L^2}$

Let $u,v \in L^2(0,T;H^1_0)$ with $u_t, v_t \in L^2(0,T;H^{-1})$. We know that $$t \mapsto (u(t), v(t))_{L^2}$$ is absolutely continuous after a change on a set of measure zero. Do we always ...
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45 views

Initial topology coincides with the locally convex topology

Suppose that $\forall j\in J: X_j $ is a locally convex space, with defining family of seminorms $(q_{jk})_{k \in K_j}$. Also let $X$ be a vector space and $T_j: X \to X_j$ a linear map $\forall j\in ...
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34 views

Riesz representation theorem, sinc functions.

There is an intimate connection between analytic functions and the completeness of sets of complex exponentials $\{e^{i\lambda_n t}\}$. If, for example, the set $\{e^{i\lambda_n t}\}$ fails to be ...
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209 views

isometric isomorphism between normed spaces and its dual

Let $E$ and $F$ be normed spaces. If $E \equiv F$ (isometry isomorphic), Does $E^* \equiv F^*$ (isometry isomorphic)? Where $E^*$ and $F^*$ are continuous dual spaces.
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24 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
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24 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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74 views

Unique unit normal vector (supporting hyperplane) of strictly convex set

When I try to prove this problem, in $\mathbb{R}^n$, we know that a set $K$ which is closed, convex, then every point $x_0\in \partial K$ admits a supporting hyperplane, i.e there exists a vector ...
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59 views

About continuity of Holder functions

I assume that the functional class $\mathcal{H}^{0}$ is defined as: In the interval $[x, x + \epsilon]$ it is true that $| f(x+ \epsilon) - f(x) | \leq C$ for a positive constant $C$. Then let the ...
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40 views

Direct sum decomposition of vector spaces and their tensor powers

Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus ...
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91 views

Exercise 1.19 in Brezis' Functional Analysis

I would like a clarification to an exercise in Brezis' Functional Analysis. Exercise 1.19(2) says Let $E$ be a normed vector space. Let $F:\mathbb R \rightarrow (-\infty, +\infty]$ be a convex ...
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30 views

On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
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28 views

How to compute the derivative of this functional on a manifold?

I'm a little puzzled by the following computation. Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from ...
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93 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
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30 views

Norm of a linear function on $c_{0}$

Let $c_0$ be the space of all sequences which converge to $0$. Let $f: c_0 \to\mathbb{R}$ by $$f(x) = \sum_{n=1}^\infty \frac{x_n}{n^2}$$ What is the value of $\|f\|$?. I am just getting used to the ...
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59 views

Prove that $C[a,b]$ with inner product $\langle f,g\rangle:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space.

Prove that $C[a,b]$ with inner product $<f,g>:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space. Now the norm induced by the inner product is \begin{align} ...
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60 views

How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper ...
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27 views

Some kind of fundamental lemma

I remember having read something like Let $X,Y \subset \mathbb{R}^n$ open and connected and let $u \in L^1_{loc}(X\times Y)$ with $$ \int_X \int_Y u \, \mathrm{div}_y \, \varphi \, dydx = 0$$ ...
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109 views

Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
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44 views

The use of Schauder fixed point in ladyzehskaya

The book Linear and Quasilinear equations of parabolic type gives the uniform parabolic pde theory in the literature. Ladyzhenskaya use Leray-Schauder rather than Schauder fixed point theorem. why? ...
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38 views

Extending linear continuous functions.

Let $E$, $F$ be normed vector spaces and $M$ a subspace of $E$. I'm trying to find an example of a function $f:M\to F$ such that $f$ is linear and continuous but that you can't extend it to ...
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23 views

A question on $ L^1(\mathbb{R})$.

Question- Let {$f_k$} be a sequence in $ L^1(\mathbb{R})$ such that $\sum\limits_{k=1}^\infty||f_k||_1<\infty$. Prove that the series $\sum\limits_{k=1}^\infty f_k$ converges in $L^1(\mathbb{R})$ ...
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105 views

Show that $S^{\perp \perp} \equiv (S^\perp)^\perp$ is the closure of $S$.

Suppose $S$ is a (not neccessarily closed) subspace of a Hilbert space $H$. Show that $S^{\perp \perp} \equiv (S^\perp)^\perp$ is the closure of $S$. I know that if $X\in H$, that $X^\perp$ is a ...
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33 views

Does a set of 'm' linearly independent continuous functions constitute a Hilbert Space

If I have a Sobolev space $\mathcal{H}^m[a,b]$ of functions $f : [a,b]\rightarrow\mathbb{R}$ where for all $f \in\mathcal{H}^m[a,b]$, $f$ and all derivatives up to order $m-1$ are absolutely ...
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41 views

Given $A_n : X\rightarrow Y$ linear and continuous, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ continuous?

Given $A_n : X\rightarrow B$, a linear and continuous operator between two Banach spaces, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ linear and continuous? My attempt: $A$ ...
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70 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
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106 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
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38 views

Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
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31 views

Eigenvalues that are functions

Let us have the Laplacian on a compact manifold $M$. Suppose I have some equation of the form $$-\Delta u(x) = f(x)u(x).$$ If $f \equiv c$ were a constant, this would be an eigenvalue problem ...