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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5
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1answer
164 views

Is there a nonnormal operator with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition : An operator $A \in B(H)$ is normal if $AA^{*} = A^{*}A$. Definition : The spectrum $\sigma(A)$ of $A \in B(H)$, is the set ...
3
votes
2answers
168 views

Hahn-Banach Theorem in the C*-algebra

What is the Hahn-Banach Theorem in the C*-algebra(or W*-algebra maybe)? If B is an nondense subalgebra of C*-algebra(or W*-algebra maybe), can we get an state f of A which is always zero at the ...
1
vote
0answers
28 views

An equivalent norm [duplicate]

Is it possible to say that $\|u\|_{H_{0}^{2}(\Omega)}$ is equivalent to $\|\Delta u\|_{2}$?
0
votes
2answers
60 views

Calculate the projection of $g(x)=\exp(−2x^2)$ onto the subspace $S$

I have problem to getting started on this one: "Let $f_1(x) = \exp(−x^2)$, $f_2(x) = xf_1(x)$, S the subspace of $L^2(\mathbb{R})$ spanned by $\{f_1,f_2\}$, and $P$ the projector onto $S$. Find $Pg$, ...
0
votes
1answer
45 views

Non-equality with Gamma functions

Let $n \in N$, $k \in Z_+$. Show that $$ \frac{\Gamma\left(k+\frac 12\right)}{\Gamma\left(k+\frac ...
2
votes
1answer
67 views

Surjectivity of a restriction map on distributions

I'm reading Kudla's exposition of Tate's thesis in the book "An Introduction to the Langlands Program" and have gotten stuck on some analytic details. Here's the setup: let $F$ be $\mathbb{R}$ or ...
6
votes
5answers
461 views

Examples of (infinite dimensional) linear operators

I'm trying to familiarize myself with linear operators. In finite dimensions it is clear to me that they are matrices. No problem there. But then in infinite dimensions matters are not so clear to me. ...
3
votes
1answer
322 views

The Banach-Mazur distance is not reached

Let $X,Y$ be isomorphic Banach spaces. The Banach-Mazur distance: $$ d(X,Y)=\inf\{\|T\| \cdot \|T^{-1}\|: T:X\longrightarrow Y \ \text{is an isomorphism} \}$$ can be rewritten as: $$ ...
1
vote
0answers
53 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
2
votes
0answers
70 views

Is there an operator such that the spectrum of all its nontrivial commutants is strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition : let $A \in B(H)$, then its spectrum $\sigma(A)$ is the set of all $\lambda \in \mathbb{C}$ such that $A - \lambda I$ is ...
4
votes
2answers
205 views

How generalize the bicommutant theorem?

Let $H$ be an infinite dimensional separable Hilbert space. Bicommutant theorem : Let $\mathcal{S}$ be $*$-subset of $B(H)$, then $\mathcal{S}''$ is the strong closure $\overline{\langle ...
2
votes
1answer
126 views

Is there an operator whose non-zero commutants are always injective?

Let $H$ be an infinite dimensional separable Hilbert space. Is there an operator $T \in B(H)$ such that, if $TA=AT$ with $0 \ne A \in B(H)$, then $A$ injective ? Bonus question : what is ...
2
votes
1answer
55 views

Different notions of differentiability

The following is somewhat unclear to me. Let $X$, $Y$ be locally convex vector spaces, let $f: X \supseteq U \longrightarrow Y$ be a (nonlinear) continuous map. Then one can say that $f$ is $C^1$ if ...
4
votes
1answer
187 views

Is there a non-locally compact Hausdorff space in which all infinite compact sets (of which there is at least one) have uncountable interiors?

Here is the background material from which I am working: The Cantor set is an uncountable compact Hausdorff subspace of $\mathbb{R}$ with empty interior. In a locally compact Hausdorff space with no ...
5
votes
0answers
162 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
5
votes
2answers
506 views

Weighted Poincare Inequality

I'm trying to prove a result I found in a paper, and I think I'm being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have ...
4
votes
0answers
90 views

An element of $\ell^2$ wanted

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is ...
3
votes
2answers
173 views

Do maps between topological spaces somehow induce maps between Banach spaces?

If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map \begin{align*} h':C_b(X)\rightarrow C_b(Y) \end{align*} (or in the other direction) where ...
1
vote
0answers
70 views

A question on convergence of solution of an integral equation.

In Pipkin's "A Course on Integral Equations", on page 24 problem 2, he asks us to find out whether or not iteration will converge uniformly for an integral equation of the second kind, i.e $u=f+Ku$ on ...
5
votes
1answer
610 views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
6
votes
1answer
111 views

Is there a non-locally compact Hausdorff space in which all infinite compact sets (of which there is at least one) have nonempty interior?

Here is the background material from which I am working: The Cantor set is an uncountable compact Hausdorff space with empty interior. In a locally compact Hausdorff space, each countable set has ...
6
votes
1answer
175 views

Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $\|xx^{\#}\|=\|x\|^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
2
votes
0answers
90 views

What is $\int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \; $?

Given $F[u]$ and $G[v]$ are functionals of a real-valued function, what is $$ \int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \quad ? $$ I have encountered such expressions for ...
1
vote
2answers
260 views

Is there an injective operator with a dense nonclosed range?

Let $H$ be an infinite dimensional separable Hilbert space. Is there an operators $A \in B(H)$ such that $Im(A) \subsetneq \overline{Im(A)} = H$ and $Ker(A) = \{0\}$ ? Bonus : We can build ...
1
vote
1answer
44 views

Linear homeomorphisms mapping an orthonormal basis into another orthonormal basis

Consider $L^2(A)$ and $L^2(B)$. If $\{a_i\}$ is an o.n basis of $L^2(A)$, how many linear homeomorphisms $F:L^2(A) \to L^2(B)$ do there exist such that $Fa_i$ is an orthonormal basis of $L^2(B)$? Is ...
7
votes
4answers
260 views

What makes irreducible representations nice?

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation. What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations ...
2
votes
1answer
229 views

Maximize a functional

Please help me how to deal with maximization of functional like this: $$F\{a(s)\} = \int\limits_0^t \left( g(a(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right)$$ where $g(x) = x ...
1
vote
1answer
62 views

Fourier series convergence in $L^2$

Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit. I was thinking ...
0
votes
3answers
36 views

Show that $f \in c_0^*$ and $||f||=\sum_{j=1}^{\infty} \frac{1}{j!}$

Let $$\begin{eqnarray} f: c_0 & \to & \mathbb{R}\\ (x_i)_1^{\infty} & \to & \displaystyle \sum_{j=1}^{\infty} \frac{x_j}{j!}\\ \end{eqnarray}$$ Show that $f \in c_0^*$ and ...
0
votes
1answer
47 views

Show that we can use a sequence $(\phi_n)$ to induce continuous functionals

Let $(\phi_n) \subset C[-1,1]$. Show that we can use a sequence $(\phi_n)$ to induce continuous functionals in $C[-1,1]$ where the dual pair is $$\langle \phi_n, f \rangle=\int_{-1}^{1} ...
2
votes
1answer
344 views

Invertibility of a linear operator on a Hilbert space.

Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such ...
1
vote
1answer
48 views

Restriction of a lower semi-continuous functional again lower semi-continuous?

Let $F: [a,b]\times \mathbb R \times \mathbb R \rightarrow \mathbb R$ be continuous, $J(u) = \int_{[a,b]} F(x, u, u') dx$ be a functional over $W^{k,p}([a,b])$. We assume that for any uniformly ...
2
votes
1answer
50 views

Restriction to $\mathbb{R}^{d-1}$ as an operator on $L^2(\mathbb{R}^d)$

Identify $\mathbb{R}^{d-1}$ with $\mathbb{R}^{d-1}\times \{0\}\subseteq \mathbb{R}^d$. Is there a bounded operator $T: L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^{d-1})$ such that $T(\phi)=\phi ...
0
votes
2answers
55 views

Continuous function space separablity

We know that $C[0, \infty)$ is complete metric space with sup norm. Is it also seperable? How to show it? Thank you.
2
votes
2answers
113 views

Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
7
votes
1answer
265 views

Fixed Point Theorems

Theorem 1. Let $B=\{x\in \mathbb R^n :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^n$ . Any continuous function $f:B\rightarrow B$ has a fixed point. Theorem 2. Let $X$ be a finite dimensional ...
3
votes
1answer
95 views

A certain Hilbert space projection operator; verification needed

Let $V \subset H$ be separable Hilbert spaces with dense and continuous embedding. For each $n$, let $V_n$ and $H_n$ be finite-dimensional subspaces of $V$ and $H$ respectively with dimension $n$. ...
5
votes
2answers
306 views

extreme points of the unit ball of the Schatten classes?

Suppose $1<p<\infty$. What are the extreme points of the unit ball of the Schatten classes $S^p$? See below for the definition of $S^p$: http://en.wikipedia.org/wiki/Schatten_norm
1
vote
1answer
282 views

Showing a certain sequence forms an orthogonal basis in $L^2$

We want to show that sequence ($\frac{1}{\sqrt{2\pi}}$,$\frac{1}{\sqrt{\pi}}\sin(x)$,$\frac{1}{\sqrt{\pi}}\cos(x)$, $\frac{1}{\sqrt{\pi}}\sin(2x)$,$\frac{1}{\sqrt{\pi}}\cos(2x)$, ...
1
vote
1answer
273 views

Norm of integral operator

Consider the operator $T(f(t)) = \int_0^t f(s)ds$, where $t \in [0,1]$, and $f(t) \in C[0,1]$. To prove $$\|T^n\| = \frac{1}{n!}$$ Thanks for suggestions.
1
vote
1answer
101 views

Mergelyan's theorem from Runge's theorem?

From Conway, A course in functional analysis, page 85. Corollary 8.5. I want to ask for a hint how to deduce Mergelyan's theorem from Runge's theorem, assuming a functional analysis rhetoric proof. ...
3
votes
1answer
850 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
8
votes
2answers
386 views

Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any ...
2
votes
2answers
124 views

How do I show that this mapping is Lipschitz continuous

Let $X$ be a Banach space. Since metric spaces are paracompact we know that $X$ is too. Then given the following cover of $X$: \begin{equation} \mathcal{N}=\{N_x\ |\ x\in X\}, \end{equation} where ...
2
votes
1answer
114 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
0
votes
2answers
66 views

Given a fourier series in $L^2$ and using it to determine a particular integral

Suppose $g \in L^2 (-\pi,\pi)$ has Fourier series is $b_0 + \sum_{n=1}^\infty (b_n\cos(nx)+c_n\sin(nx))$. From this we want to determine what $\frac{1}{2\pi}\int_{-\pi}^{\pi}|g(x)|^2dx$ equals. I ...
2
votes
1answer
66 views

An imbedding question

Is it possible to say that $$ H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\hookrightarrow H_{0}^{1}(\Omega). $$ Precisely, I am dealing with this question: Is it possible to have the following estimate if we ...
5
votes
1answer
437 views

What is an example of a bounded, discontinuous linear operator between topological vector spaces?

I am thinking there might be an example between the space of compactly supported smooth functions on the real line (chosen because it is non-metrizable under the standard topology for this space of ...
2
votes
1answer
121 views

Want to show an operator is compact

With $V=L^2(0,T;H^1(\Omega))$, let $A:V \to V^*$ with $$\langle Au,v \rangle = \int_0^T \int_{\Omega} \nabla u(t) \cdot \nabla v(t).$$ I want to show that $A$ is a compact operator. So, one way to ...
1
vote
1answer
148 views

Is tensor product of Sobolev spaces dense?

My question is: is $W_2^k(\mathbb{R})\otimes W_2^k(\mathbb{R})$ dense in $W_2^k(\mathbb{R}^2)$, and more generally is this true in $\mathbb{R}^d$? I found this post: Tensor products of functions ...