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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32 views

Is a function of bounded semivariation bounded?

Let $X$ be a topological vector space and let $f:[a,b]\to X$. We say that $f$ is of bounded semi-variation in $[a,b]$ if the set $SV(f,[a,b])$ consisting of all the elements of the form $$\sum_{i=1}^n ...
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1answer
56 views

Holomorphic functions on the product of open sets.

Is it true that $$ \mathcal H(\mathrm U \times \mathrm V) \simeq \mathcal H(\mathrm U) \widehat{\otimes} \mathcal H(\mathrm V) $$ for open two open affine sets $\mathrm U$ and $\mathrm V$? Edit: I ...
8
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1answer
190 views

Ideals in $B(H)$ are self-adjoint

It is known that every (closed two-sided) ideal in a $C^{*}$-algebra is self-adjoint. The proofs that I've seen involve functional calculus and approximate units. I am wondering whether there is a ...
3
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1answer
42 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
2
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1answer
71 views

Operator's norm

Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum and $T^{-1}$ is bounded such that $$\|Tf\| \leq M \|Lf\|^{a}\|f\|^{1-a}, ...
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2answers
153 views

a problem about Lp space and Holder inequality [duplicate]

If for any $g$ in $L^q$, $fg$ is in $L^1$, prove that $f$ is in $L^p$. $p>1$, $p$ and $q$ are conjugate. It is sort of an inverse version of Holder inequality. We are considering Lebesgue measure ...
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2answers
67 views

Prove a set is closed using continuous function

Suppose $U$ is an isometry isomorphism from $C(Q)$ to $C(K)$ where $C(Q)$ is the Banach space which contains all real value continuous function defined on $Q$ and its norm is the sup-norm, i.e. ...
3
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1answer
135 views

A question on the spectral projection

I am reading a paper about spectral theory. And I meet with some problems. An operator $K\in L(X)$ is said to be algebraic if there exists a non-trivial complex polynomial $h$ such that $h(K)=0$. By ...
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3answers
108 views

Prove a space is Hilbert [duplicate]

I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ...
4
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3answers
189 views

Continuous Linear Functional on $\ell^{\infty}$

I'd like help answering two questions. 1) Prove that there is a continuous linear functional on $\ell^\infty$ such that $f(e_n)=0 \ \forall n \in \Bbb{N}$ and $f(a)=5$ where $a=(1,1,1,1,1,1,\ldots)$. ...
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0answers
58 views

Continuation principle

In the following $g$ is a function related with some norms of solution of a certain differential eqution: $g$ be a nonnegative continous (if necessary, it is monotone increasing) function satisfying ...
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1answer
65 views

interpretation of a lemma

In this book, chapter $2$, lemma $2.2.2$, states the following: Suppose $K$ is a countably compact space with $f,g \in C(K)$ and $\|f\| \leq 1$ and $\|g\| \leq 1$. Then $f \in st(g)$ if and only ...
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1answer
248 views

Bessel's inequality implying convergence

Define Bessel's inequality $$ \sum_{n=1}^N|a_n|^2\leq \|x\|^2 $$ where $a_n=\langle x,e_n \rangle$. Lemma Let $\{e_n\}_{n\geq 0}\subset H$ be a complete orthonormal set. If $x\in H$ and ...
0
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1answer
136 views

Baire's theorem proof

> A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets. I don't understand the following point the proof: we assume that $X$ is the countable union of ...
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1answer
111 views

Total variation for functions, Meaning of supremum as used here?

On Wikipedia article, here: http://en.wikipedia.org/wiki/Total_variation, on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a ...
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1answer
147 views

Gelfand triple for tensor product of Hilbert spaces

Is there any dense embeding $\to$ that makes $H^1_0(D) \otimes L^2(\Gamma) \to L^2(D) \otimes L^2(\Gamma) \to (H^1_0(D) \otimes L^2(\Gamma))^{*}$ a Gelfand tripe? In fact we may only answere to the ...
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1answer
43 views

Show that the imbedding $C^{m+1}(\overline{\Omega}) \to C^{m,1}(\overline{\Omega})$ is not compact

Let $\Omega \subseteq \mathbb{R}^n$ be open. Let $C^m (\overline{\Omega})$ be the Banach space of functions such that each partial derivative $D^{\alpha}f$, $|\alpha| \le m,$ exists and is uniformly ...
4
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1answer
109 views

A use of Hahn-Banach and Riesz Representation

Let $X$ be a compact Hausdorff topological space. Suppose $X$ is not a singleton set and $C(X)$ denotes the space of continuous functions on $X$. Do we have that for all $L \subset C(X)$ a nondense ...
6
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1answer
301 views

equivalent? algebraic definition of a partial isometry in a C*-algebra

An element $a\in\mathfrak{A}$ (unital C*-algebra) is a partial isometry if $a^*\cdot a $ is projection. Can one recover the equivalent caracterizations of a partial isometry in ...
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0answers
45 views

Particular integral equation

Let $a,\sigma, n>0$ be some parameters and define the conditional probability density function $$ p(x,y):= \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-n-x)^2}{2\sigma^2}\right). $$ Is it ...
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1answer
140 views

Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and \begin{equation} ...
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0answers
157 views

Conditional expectation on the space of bounded linear operators

In the paper from the link http://arxiv.org/pdf/0906.0139.pdf the author uses a diagonal conditional expectation. We take a seperable Hilbert space $H$ and fix an orthonormal basis $(e_n)_{n \in ...
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0answers
49 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
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2answers
85 views

Just a sanity check in basic functional analysis

Consider the algebra $C(S^1)$ of continuous functions $S^1 \to \mathbb C$ together with the $\|\cdot\|_\infty$ ($\sup$-norm). I am thinking that: (?) The (sub-)algebra generated by $\rm{id}$ and ...
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2answers
112 views

Let $z_1 = x_1 +iy_1 $ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $< z_1 , z_2> = x_1x_2+y_1y_2$

Let $z_1 = x_1 +iy_1$ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $\langle z_1 , z_2 \rangle = x_1x_2+y_1y_2$ For non zero $z_1$ and $z_2$ prove ...
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96 views

Let V be an inner product space. If $ x⊥y $, then show that

Let $V$ be an inner product space. If $x_i ⊥ x_j $ when $i\neq j$, then show that $$\Bigg\Vert\sum_{i=0}^n x_i\Bigg\Vert^2\ =\sum_{i=0}^n \Vert x_i \Vert^2. $$
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137 views

Normal compact operator commute with bounded self adjoint operator in Hilbert space.

Suppose $H$ is a Hilbert space and $A:H\rightarrow H$ is a normal compact operator such that $\ker(A)=0$. show that if $B$ is a bounded self adjoint operator that commutes with $A$ then the spaces in ...
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1answer
58 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
87 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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1answer
562 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.
2
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0answers
51 views

generalizations of continuous 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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0answers
84 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 ...
3
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1answer
60 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
120 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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1answer
96 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
48 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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1answer
387 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) = ...
0
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1answer
360 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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1answer
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 ...
1
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3answers
153 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: ...
5
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1answer
176 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
116 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 ...
1
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1answer
22 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 ...
0
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1answer
120 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
269 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
113 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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99 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
0
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1answer
75 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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2answers
69 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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2answers
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 ...