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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56 views

Different types of continuity in $\ell^2$

Consider the following functional $J$ on $\ell^2$ which for $x = \{x_n\}$ is defined by $$J(x) = \sum_{n=1}^{\infty}n^{1/n}x_{n}^{2}.$$ Is $J$ continuous? Is $J$ lower semi-continuous? Is $J$ ...
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2answers
77 views

Clarifying the definition of essential self-adjointness

If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
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318 views

Eigenvalue problem for the Laplacian on the unit ball [closed]

I want to find out what are the eigenvalues and eigenfunctions of the eigenvalue problem for the Laplacian on the unit ball in $\mathbb R^3$, with the Dirichlet boundary conditions.
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48 views

generalizations of continous operators?

What are generalizations of the notion of continous linear operator $P:X\to X$, where X is a Banach space? I'm looking for some broader class of operators that nevertheless share some properties of ...
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67 views

Calculus of variations: the inside function has an integral

It is known that if the functional $$J=\int_a^b L(x,f(x))dx \tag{1}$$ has an extremum, then the Euler equation $\frac{\partial{L}}{\partial{f(x)}}=0$ holds. My question is, for example, what if ...
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51 views

A simple question about completely positive linear maps

Let $A$ be the C*-algebra and $M_{n}(A)$ be the C*-algebra of $n\times n$ matrices with entries in $A$. We use $(a_{ij})$ to denote the element of $M_{n}(A)$. My question is: For every $a\in A$, ...
2
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2answers
96 views

Homeomorphism vs bijective and continuous

I am trying to understand the difference between a function which is a homeomorphism and for example a function $f:U \rightarrow V$ (U, V open sets) which is bijective and strictly increasing . The ...
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92 views

Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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1answer
47 views

How to prove the demicountinuity of nonlinear operators?

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$ \mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1. $$ How to prove that ...
3
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191 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
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46 views

Normed Linear Spaces.

If $X$ is a real valued function whose value at $x$ is denoted as the norm of $x$, then prove that 1). The norm of $x$ is greater than or equal to zero. 2). The norm of $x$ is not zero iff x is not ...
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220 views

Minkowski's Inequality For Infinite Sum.

Minkowski's inequality for infinite sum(where it ranges from zero to infinity), is the proof the same with the Minkowski's inequality(where the sum is finite e.g 0 to n) ? If they are not the same, ...
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65 views

An Example Of A Banach space. [closed]

Consider the linear space $\mathcal{L}_\infty$ and let $x\in\mathcal{L}_\infty$ where $x=(a_1,a_2,...,a_n)$ and taking the norm of $x$ to be $\sup x_i$. My questions are: 1). When defining $x$ in ...
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3answers
145 views

if $\|x+y\|=\|x\|+\|y\|$, then $\|\alpha x+\beta y\|=\alpha \|x\|+\beta \|y\|$

Let $X$ be a normed linear space. Assume that for $x,y \in X$, we have $||x+y||=||x||+||y||$. Show that $||\alpha x+\beta y||=\alpha ||x||+\beta ||y||$ for every $\alpha,\beta \geq 0$. My attempt: ...
6
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124 views

Weak and almost everywhere convergence

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that $p\in (1,\infty)$. Assume that the sequence $u_n\in L^p(\Omega)$ satisfies: There is $u,w\in L^p(\Omega)$ such that $u_n\to u$ a.e. ...
2
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1answer
93 views

Arzela-Ascoli and adjoint of compact operator compact

I have seen in this thread a nice answer where it is shown that Thread that the adjoint operator of a compact operator is compact by using the Arzela Ascoli theorem. Unfortunately, there is one thing ...
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1answer
20 views

Descomposition on temporal sobolev space

Let $\Omega$ an open subset of $\mathbb{R}^2$ with Lipschitz boundary. Can I descompose in a unique way any $u\in L^2(0,T;L^2(\Omega))$ such that for all $t\in [0,T]$, $u(t)=u_1(t)+u_2(t)$ with ...
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1answer
102 views

Partial derivative w.r.t an integration

For example, I have a functional $$J(f)=\int \frac{f(x)}{1+x^2}dx.$$ How to calculate $\frac{\partial J}{\partial f(x)}$? Does it equal to $\int \frac{1}{1+x^2}dx$? It seems that the question is ...
2
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1answer
185 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
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1answer
81 views

Show that the following functional is Frechet differentiable in Hilbert space

I need to show that the following functional is Frechet differentiable: $$ f(u) = \|u\|^2_{H} \ \ \text{in a real Hilbert space} \ \ H $$ Solution: As far as I understand, I need to take a Taylor ...
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84 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
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1answer
60 views

closed subspace $Y$ implies existence of non-zero linear functional $g$ such that $Y \subset \ker(g)$

I am working on an exercise and I am not sure if I am on the right track, so if anyone could give some hints I would be grateful. The exercise is If $Y$ is a proper closed subspace of $X$, prove that ...
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63 views

Does there exist a function in $L^1$ such that $u * f = f$ for all f in $L^1$

While studying for exams, a practice question came up which is Does there exist a function $u \in L^1({\mathbb{R}^d})$ such that $u * f = f$ for all $f \in L^1({\mathbb{R}^d})$? I was thinking ...
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35 views

An exercise about compact operater

If $A \in \mathfrak{B}(H)$ and $H$ is a Hilbert space, $AT=TA$ for every compact operater $T$, show that $A$ is a multiple of the identity operater. I don't what is "multiple of the identity ...
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1answer
222 views

The adjoint of finite rank operater is finite rank

If $T \in \mathfrak{B}_{00}(\mathfrak{H},\mathfrak{K})$, show that $T^{*} \in \mathfrak{B}_{00}(\mathfrak{K},\mathfrak{H})$ and $dim(ran T) = dim(ran T^{*})$. The ...
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1answer
75 views

Convex symmetric set in real vector space is balance

Show that a convex set in a real vector space is symmetric if and only if the set is balance. For the backward direction, i.e. if the convex set is balance in real vector space, then it is symmetric ...
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40 views

Prove that a give sequence of function is a base of $L^2([0,1])$

Consider $(\phi_k)_{k \geq0} \in \mathcal C^{\infty}([0,1])$ with $\phi_k \not\equiv 0 $ such that $$\int_0^1 \phi_k(s) ds = 0, \quad \forall k\geq 1$$ and $$\sup_{ t \in [0,1]} \left | \frac{d}{dt} ...
2
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1answer
114 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
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1answer
94 views

Show that the following functional is Frechet differentiable

I am new to this and I need to show that the following functional is Frechet differentiable: \begin{equation} f(u) = \sin(u(1)) \ \ in \ \ C[0,1] \end{equation} What I have already done: ...
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0answers
72 views

Complex Power of a differential operator

Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum $$ S_N = \sum^N_{n = 1} B_n ...
4
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3answers
92 views

$A^2$ self-adjoint and Compact, prove $A$ has an eigenvalue

Suppose $H$ is a Hilbert space and $A \in L(H)$ is such that $A^2$ is compact and self-adjoint. Prove that $A$ has an eigenvalue. (Here $L(H)$ is the set of bounded linear operators on a Hilbert ...
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0answers
94 views

Variants of the bump function.

The title of this question isn't really clear because of the 150 char limit. What I actually want to ask is this: If I would have a bump function for $-1 < x < 1$ and I would have some ...
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1answer
64 views

Determinant inequality for trace class operator

Let $A$ be a trace class operator on a Hilbert space. I wonder if there is an estimate of the form $$ |\log \det (I + A)| \le C\|A\|_1, $$ for some constant $C$, where the norm on the right is the ...
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0answers
61 views

Balanced subset of a vector space

Suppose $A$ is a balanced subset of a vector space $V$. Show that $[0,a] \subset A$ for all $a \in A$. By definition, $[0,a]=\lbrace (1-\lambda)a:a \in A, \lambda \in [0,1]\rbrace$. Also by ...
3
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1answer
132 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
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1answer
54 views

A problem about projective operater

Let $P$ and $Q$ be projective on a Hilbert space $H$. Show that $P+Q$ is projective if and only if $\mbox{ran }P \perp \mbox{ran }Q$. The sufficiency is easy. About the necessity, suppose $P+Q$ is ...
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2answers
117 views

Functional Analysis, operator theory, eigenvalues of a operator

We have $$T_\alpha:C[a,b]\to C[a,b]$$ $$T_\alpha f= \alpha f$$ where $C[a,b]=\{ f:[a,b]\to \mathbb{R} \quad f$ is continuous} and $\alpha\in C[a,b]$ fixed. Show: Spectrum of $T_\alpha\equiv ...
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149 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
3
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0answers
72 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
3
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1answer
119 views

Question about projections on Hilbert space

Let $P_i$ be projections from a Hilbert space $\cal{H}$ to its closed subspace $\cal{H}_i$, $i=1,2,\cdots,n$, such that $\sum^n_{i=1} P_i$ is also a projection. And let $P$ be a projection from ...
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1answer
52 views

What is the name of this object?

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let $$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$ ...
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1answer
32 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
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1answer
48 views

Extension of a zero linear functional

How can I show using Hahn-Banach theorem that, if $E$ is a real vector space, $F$ is a proper vector subspace of $E$, and f is the zero linear functional $f:F\to\mathbb{R}$ such that $f(x)=0$ $\forall ...
2
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2answers
129 views

Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
2
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1answer
88 views

Find the norm of $A$ where $(Af)(t)=tf(t)$

I have the following problem that I would like to ask you about: I have $X$ as my normed linear vector space and $B(X,X)=B(X)$ as my space of all operators $A: X \to X$, where for all $A \in B(X)$ is ...
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1answer
63 views

Confusion over an example of weak limit in L2

I understand the definition of weak limit, but I'm confused about the weak $L^2$ limit of the sequence $g_n = n1_{[0,1/n^2]}$. The $L^2$-norm of each of these functions is $1$. Does this mean the ...
4
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0answers
83 views

Linear dimension of banach spaces

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $||x||=|x_1|+...+|x_n|$. ...
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2answers
51 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
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1answer
58 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
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1answer
277 views

Hahn–Banach Theorem for Normed Spaces: not unique extension

Let $\ell^{\infty}$ be the set of bounded sequences in $\mathbb{F}$, with the supremum norm. $c \subset \ell^{\infty}$ the sequences whose limit exists. Then there exists a $f \in ...