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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1answer
300 views

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
1
vote
1answer
70 views

What is the $C^1$ domain?

I just came across the Deuflhard & Bornemann text on Scientific Computing with ODEs where they write, for example: $f \in C(\Omega, \mathbb{R}^d)$ In other places they use ...
2
votes
1answer
395 views

How is an integral with respect to a Hausdorff measure defined?

In a reply by Corey: For integrals of scalar-valued functions on unoriented subsets of $\mathbb{R}^n$, one can use the Lebesgue integral with respect to $k$-dimensional Hausdorff measure ...
1
vote
1answer
174 views

When is a compact operator differentiable?

When is it possible to prove that a compact operator $T: V \to V$ where $V$ is a Banach space is also differentiable? Fréchet differentiable? PS: There is a further information which might help. My ...
2
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0answers
81 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
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0answers
340 views

Weak derivative

Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative? Is the mean value theorem for integration ...
2
votes
2answers
244 views

reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
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0answers
109 views

How can I compare unbounded linear operators?

Let $X$, $Y$ be Hilbert spaces. Let $S, T : X \rightarrow Y$ be unbounded operator. Suppose $S$ and $T$ be bounded operators. Then we can compare by their maximum distance on the unit ball of $X$. ...
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1answer
1k views

Dual Space of Convergent Sequences

For every linear functional $T$ on the space of convergent sequences in $\mathbb{R}$, how can I show it can be expressed $T(\{s_{n}\}) = \sum_{n \in \mathbb{N}} s_{n}T(e_{n})$ where $e_{n}$ are the ...
3
votes
0answers
393 views

Modification of Mazur's lemma

Working through Brézis and solving an exercise I have a question about my solution. It's well known that, if $x_n$ converges weakly to $ x $ in a Banach space $X$, then there exists a sequence ...
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2answers
240 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...
3
votes
1answer
64 views

Algebraic description of coisometries

Let $H$ be a Hilbert space. An operator $T\in\mathcal{B}(H)$ is called coisometric if it maps open unit ball of $H$ onto open unit ball of $H$. Please tell me how to prove that condition $T$ is ...
4
votes
2answers
363 views

Boundedness of operator on Hilbert space

I have the following question: let $\mathcal{H}$ be a Hilbert space and $\{\varphi_{i}\}_{i \in \mathbb{N}}$ be an orthonormal basis. Furthermore let $T: \mathcal{H} \rightarrow \mathcal{H}$ be an ...
1
vote
1answer
124 views

Complicated “functional integral”

I came across the following "functional" at work: $$ \Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda $$ it's part of an optimization problem that tries to find ...
7
votes
2answers
512 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: ...
3
votes
0answers
148 views

Find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$

I need to find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$ in $L_2[0,\pi]$. I know that this operator is self-adjoint, so its residual spectrum ...
2
votes
1answer
152 views

Non compact embedding

Could someone please explain to me, why the embedding $\iota \colon C^0( \overline{\Omega}) \to L^2(\Omega)$ is not compact? $\Omega=(0,1)$ and $ \overline{\Omega}$ denotes the closure of $\Omega$. ...
3
votes
1answer
205 views

Dual space $E^*$ metrizable iff E has a countable basis

I have trouble proving the following theorem: If $E$ is a locally convex, Hausdorff topological vector space, then $E^*$ is metrizable if and only if $E$ has an (at most) countable basis. I've ...
3
votes
2answers
1k views

Definition of Compact Mapping

I was reading around the other day and came across the term "compact mapping". After googling, I saw the following two definitions: Let $X$ be a topological space. Then a mapping $f:X \to X$ is ...
11
votes
1answer
547 views

About example of two function which convolution is discontinuous on the “big” set of points

I want to ask about example of real valued functions defined on the real line such that their convolution exist in every point and is discontinuous on a "large" set, for example on each point of some ...
4
votes
1answer
159 views

$c_0[0,1]$ in $C(K)$

Let $K=[0,1]\times \{0,1\}$ be endowed with the topology arising from the lexicographic order on it. It is known that $K$ is compact, Hausdorff, first-countable and perfectly normal. Furthermore, the ...
6
votes
2answers
261 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
2
votes
0answers
136 views

Show that $(Y,\| h\|_Y)$ is a Banach Space

Let $E$ a Banach Space. Let $$Y:=\{h:E\to\mathbb{R} \ : \ h \text{ bounded, Fréchet differentiable and Lipschitz} \} .$$ Let $\|h\|_Y:=\|h\|_{\infty}+\|h'\|_{\infty}$. Show that ...
2
votes
1answer
278 views

Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is ...
2
votes
1answer
354 views

Well-posedness of the Poisson problem with mixed boundary conditions

let $\Omega \subset \mathbb R^n$ be a subdomain whit Lipschitz boundary, i.e. locally any part of the boundary looks like the graph of a Lipschitz continuous function, after some affine coordinate ...
0
votes
2answers
172 views

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$?

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$? If $L$ is a linear map between Banach spaces $V$ and $W$, why is this true? Also, is this true for $L$ not a linear map? Thanks! ...
1
vote
1answer
109 views

Necessary and sufficient condition of being dissipative

I want to know a necessary and sufficient condition on $m:\Omega \mapsto\mathbb{C}$ such that the multiplication operator $M_{m}$ is dissipative in $L^{p}(\Omega)$, where $\Omega$ is a Banach space. ...
1
vote
1answer
53 views

Proof of lower semi-continuity of $x\mapsto \|x\|_{bv}$

It is a part of the proof of lower semi-continuity of mapping $x \mapsto \|x\|_{bv}$, where $x\colon[0,T]\to E$, $(E,d)$ is a metric space, and the norm is defined as ...
2
votes
0answers
52 views

Embedding of $l^p$ space [duplicate]

Possible Duplicate: Inequality between $\ell^p$-norms First, I'll start by simply showing you the problem: Let $1 \leq p \leq q < \infty$. Determine that $l^p \subseteq l^p$ by proving ...
2
votes
1answer
231 views

Domain of closed operator and weak convergence: elementary proof and extensions?

Suppose $H,K$ are separable Hilbert spaces and $A : H \to K$ is a closed, densely defined, unbounded operator with domain $D(A)$. The following fact is often useful: Proposition. Suppose $x_n ...
4
votes
2answers
1k views

How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)

I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\ $ in $\ L_2[0,1]\ $ space. And I want to understand how to solve it, not just obtain an answer.
10
votes
4answers
363 views

Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and ...
2
votes
1answer
133 views

Reverse Uniform Boundedness Principle

I believe that what I am about to ask has a negative answer, but I can't seem to find a quick counterexample. Consider a family of bounded operators $\{T_\alpha\}_{\alpha \in \mathcal{A}} \subset ...
1
vote
0answers
245 views

Explanation for something in a Functional Analysis book by Kesavan

My wife is reading "Functional Analysis and Applications" by Kesavan. On page $140$, at the bottom it says "Let $\Omega$ be an open set, $f\in L^2(\Omega)$, and $u\in H^1_0(\Omega)$ such that ...
6
votes
1answer
199 views

Spectrum of $\int\limits_0^x f(t) dt$ operator

Let $A\colon E\to E$ definied by $A(f)(x)= \int\limits_0^x f(t) dt$. I have to find the spectrum of $A$ in the cases $E=C[0,1]$ and $E=L_2[0,1]$. I have proved that $A$ has no eigenvalues, but I can't ...
8
votes
1answer
2k views

Properties of dual spaces of sequence spaces

Can you tell me if I got the following homework right? Nitpicking is welcome. a) Recall that $$ c_0 (\mathbb{N}) = \{ f: \mathbb{N} \rightarrow \mathbb{C} \mid \lim_{n \rightarrow \infty } f(n) ...
5
votes
3answers
3k views

How to show that $C=C[0,1]$ is a Banach space

Let $C=C[0,1]$ be the space of all continuous functions on $[0,1]$. Define $\|f \|=\max \ |f(x)|$. I want to show that $C$ is a Banach space. Below is my attempt and I was wondering if it's ok. ...
1
vote
1answer
111 views

Dense subset of given space

If $E$ is a Banach space, $A$ is a subset such that $$A^{\perp}:= \{T \in E^{\ast}: T(A)=0\}=0,$$ then $$\overline{A} = E.$$ I don't why this is true. Does $E$ has to be Banach? Thanks
5
votes
1answer
244 views

Banach-Steinhaus theorem for nets?

Consider an uncountable set $I$ and let $A=\mbox{Fin}(I)$ be the family of finite subsets of $I$ ordered by inclusion. Let $E$ be a normed space and $F$ be a Banach space. Suppose moreover we have a ...
2
votes
0answers
80 views

Prove $\forall$ compact $M:\ M \subset C\quad \exists A:l_2\rightarrow l_2, \sigma(A)=M$ [duplicate]

Possible Duplicate: Operator whose spectrum is given compact set Can spectrum “specify” an operator? Prove that for each nonempty $M$ - compact subset of $\mathbf{C}$ exists ...
6
votes
2answers
406 views

Modulus of Continuity, Take 2

This is a follow-up to my last question: Modulus of Continuity. I accidentally asked the wrong question there, so I'm going to start over and hopefully ask the right question. I'll repeat the ...
4
votes
2answers
304 views

The spectrum of normal operators in $C^*$-algebras

Suppose that $A$ is an infinite-dimensional $C^*$-algebra. Is it true that there must be a normal element with non-discrete spectrum? If that is not true must there at least be a normal element with ...
3
votes
1answer
559 views

Interpolating between the $L^{p}$ norm and the Hölder semi-norm

Set-up. For a bounded continuous function $u \colon \mathbb{R}^n \to \mathbb{R}$, the $\gamma$-Hölder semi-norm of $u$ is $$ \begin{eqnarray} [u]_{C^\gamma} &=& \sup \left\{\frac{|u(x) ...
1
vote
0answers
665 views

cokernel and kernel of adjoint operator

Let $L$ be a linear operator from Banach space $X$ to $Y$. Is the dimension of the kernel of the adjoint of $L$ the same as the dimension of the cokernel? The cokernel is $Y/(Im L)$. Also, is the ...
6
votes
2answers
866 views

Modulus of Continuity

Let $\rho(t)$ be a function on the set $\mathbb{R}^+$ of nonnegative real numbers such that: $\rho$ is nondecreasing (and continuous - thanks for the correction) $\rho(t) = 0$ if and only if $t = 0$ ...
1
vote
1answer
83 views

self linear operator

I'm trying to solve this question on self-adjoint in complex numbers. I'm stuck and need help. Consider $ L =\frac{d^3}{dx^{3}}$ to be a linear operator which acts on a function from $[0,1]$ to $\Bbb ...
4
votes
2answers
466 views

A counterexample to theorem about orthogonal projection

Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other ...
3
votes
2answers
190 views

Distance of functions defined on a Hilbert Space

In our Topology class, we touched on Hilbert spaces for a couple of weeks. I've been studying various problems around the topics we covered, and I came across this one on a list of supplemental ...
10
votes
2answers
3k views

Proof: $X^\ast$ separable $\implies X$ separable

Can someone tell me if I got the following right: Assume $X$ to be a normed vector space over $\mathbb{R}$. Prove that if the dual space $X^\ast$ is separable then $X$ is separable as well. I'm ...
2
votes
1answer
421 views

Relation between two orthogonal projections in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and let $P$ and $Q$ be two orthogonal projections to closed subspaces $M$ and $N$ respectively. Prove that: If $PQ$ is an orthogonal projection then it's range ...