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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7
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2answers
340 views

Fourier Transform of $\ln(f(t))$

I want to compute Fourier transform of $\ln(f(t))$ maybe in a sense of distributions? Where we can assume that: $f(t) > 0$ $f(t) \in L^1$ $f(t)$ is continuous $\lim_{t \to \infty} f(t)=0$ and ...
2
votes
1answer
69 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
4
votes
1answer
88 views

Existence of regular Borel measure

Let $X$ be a $\sigma$-compact and locally compact space, and let $\Lambda:C(X)\rightarrow \mathbb{C}$ be a linear functional such that $\Lambda(f)\ge0$ if $f\ge0$. How to show that exist exactly one ...
2
votes
2answers
62 views

Fock Space: NESS

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
2
votes
1answer
49 views

How to prove $\mathcal{L}e^{-U}g = e^{-U}\mathcal{L}^*g$ for Fokker-Planck operators

Let $U$ be sufficiently smooth, $\beta$ a constant and $$ \mathcal{L}p = \frac{1}{\beta}\Delta p + \nabla\cdot(p\nabla U)\\ \mathcal{L}^*g = \frac{1}{\beta}\Delta g - \nabla g \cdot\nabla U. $$ Now ...
1
vote
2answers
79 views

From continuous to bounded Borel functions

I know that we can extend the functional calculus of bounded self-adjoint operators to bounded Borel functions. I want to do the same for unbounded self-adjouint operators. Therefore assume that $T$ ...
0
votes
2answers
22 views

Question about orthogonal projection

Let $\mathcal{H}$ be a Hilbert space. I am trying to show that every self-adjoint idempotent continuous linear transformation is the orthogonal projection onto some closed subspace of $\mathcal{H}$. ...
2
votes
1answer
49 views

Why does this completion of a Sobolev space contain constant functions? Please explain text.

Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to ...
1
vote
2answers
46 views

Fock Space: Formal Adjoints

Problem Given a pre-Hilbert space $\mathcal{H}$. Consider unbounded operators: $$S,T:\mathcal{H}\to\mathcal{H}$$ Suppose they're formal adjoints: $$\langle ...
1
vote
1answer
43 views

Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
3
votes
1answer
48 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
7
votes
0answers
98 views

Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
1
vote
0answers
31 views

Showing map is isometry between Banach quotient space

I have a closed subspace $Y$ of a Banach space $X$ and a map $T: X'/Y^{\circ} \to Y'$ given by $[f] \to f|_y$. The norm in $X'/Y^\circ$ is given by $\|[f]\| = \inf \{ \|f-h\| : h \in Y^\circ \}$. I'm ...
1
vote
0answers
41 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
1
vote
2answers
87 views

Extending linear functional in non-unique way

I'm trying to find an example of when the extension of a functional in the Hahn-Banach theorem is not necessarily unique. I'm looking at the space of continuous functions on $[0,1]$ and I'm trying to ...
2
votes
1answer
85 views

Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
6
votes
1answer
105 views

Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$

I'm trying to show that if $f$ is a continuous function on $[0,1]$ and $\int_0^{1} f(x)e^{nx}\,{\rm d}x = 0$ for all $n = 0, 1, 2, \dots$, then $f(x) = 0$. I'd like to use Weierstrass approximation ...
1
vote
1answer
102 views

Continuous operator between Banach spaces, closed range

I have some problems proving the following: $T: X \rightarrow Y$ is a continuous, linear operator between Banach spaces. Prove that $T$ is surjective $\iff$ $T^* : Y^* \rightarrow X^*$ is injective ...
1
vote
1answer
24 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
2
votes
1answer
79 views

Use Poisson summation formula to prove Gaussian sum formula

The Poisson summation formula states that for any Schwartz function $f$, $\sum\limits_{k\in\mathbb{Z}}f(k)=\sum\limits_{k\in\mathbb{Z}}\hat{f}(k)$, where $\hat{f}$ is the Fourier transform of $f$. The ...
1
vote
0answers
40 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
1
vote
1answer
40 views

Can the equivalence of isometry and unitarity of a linear operator be extended to infinite dimensions?

I was wondering whether it is possible to extend the following standard theorem to infinite dimensional Hilbert spaces? Let $M\in \mathbb{C}^{n\times n}$ arbitrary. The matrix $M$ is unitary if and ...
3
votes
1answer
43 views

Conway base 13 function, Darboux functions and unbounded linear functionals

After reading many interesting posts here about Darboux functions and Conway's base 13 function (http://en.wikipedia.org/wiki/Conway_base_13_function) I have some questions that I don't seem to answer ...
0
votes
1answer
56 views

Dilating sinusoidal period at certain abscissa $x$?

Lets take a look at this function : $$f(x) = \sin\left(\frac{\pi x }{2}\right)$$ when $x$ tends to $1$ this functions get closer to $1$ by bigger values now look at this one : ...
0
votes
1answer
15 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
0
votes
0answers
62 views

Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
1
vote
1answer
50 views

Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
2
votes
1answer
45 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
1
vote
1answer
61 views

An exercise showing that $l^1$ is not the dual of $l^\infty$

there is a well known fact that $l^1$ is not the dual of $l^\infty$. An exercise Folland's Real analysis serves as an example for this.(Page 192 ex 19) Define $\phi_n \in (l^\infty)^*$ by ...
0
votes
0answers
91 views

When $1 \le p \le \infty, p\ne 2$, $L^p$ space is not a Hilbert space

It suffices to show that when $1 \le p \le \infty, p\ne 2$, $L^p$ norm does not arise from an inner product.(there is a hint saying that we can use the parallelogram law) I can proof a special case ...
1
vote
1answer
51 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
1
vote
1answer
40 views

Special integrands in the calculus of variations

Most techniques in the calculus of variations that I know of, deal with integrands of the form $W(x, \phi(x), \nabla \phi(x)): \Omega \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \to ...
0
votes
1answer
55 views

Density of continuosly differentiable function in space of continuous functions

Let $C([0,1])$ be the set of all real continuous functions with the standard supremum norm. Let $C^1([0,1])$ be the set of all real continuosly differentiable functions on $(0,1)$ such that the ...
0
votes
0answers
22 views

Approximating the weak gradient of the constant function in $L^p$

I want to find a sequence $u_n:(0,\infty) \to \mathbb{R}$ such that $u_n \to k$ pointwise and $\nabla u_n \to 0$ in $L^p$, where $k$ is the constant function equal to $k \in \mathbb{R}.$ Will the ...
2
votes
2answers
54 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
1
vote
1answer
73 views

Completion of a Banach space with respect to a different norm

Let $(X,|\cdot|_X)$ be a Banach space. Define a space $Y$ as the completion of $X$ under a norm $$|u|_Y = |u|_X + |Tu|_Z$$ where $T:X \to Z$ is a linear continuous map where $X \subset Z$ is a ...
0
votes
1answer
59 views

A Problem on Locally Convex Spaces

In the book A Course in Functional Analysis by Conway, there is the following problem: Problem. Let $ X $ be a completely regular topological space, and let $ C(X) $ denote the set of all ...
1
vote
0answers
21 views

Does the convolution property suffice to show $\hat \chi*\hat \chi=\hat \chi$?

Given a compact and sufficiently regular set $\Omega\subset\mathbb{R^n}$ and its characteristic function $\chi=\chi_\Omega$, I would like to conclude that the (inverse) Fourier transform ...
1
vote
0answers
18 views

Eigenelements of Neumann Laplacian satisfy $\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$

Let the eigenvalues of Neumann Laplacian on a bounded open domain be given by $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 ...$ associated to eigenfunctions $\varphi_0, \varphi_1, ...$. Let $u \in ...
0
votes
0answers
28 views

Detail in a proof about energy minimizing harmonic maps

Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional ...
0
votes
1answer
42 views

Prove that if ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ then $f = \sum_{j}<f, \phi_j> \phi_j$

Let $\phi_k$ be some sequence of real functions in an infinite Hilbert space $H$ such that there exists $A \in \mathbb{R}$ such that for all $f \in H$ ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ ...
1
vote
1answer
28 views

Weak convergence in $BC(\mathbb R^+;X)$

I know that a sequence $u_n\in C([a,b];X)$ ($X$ is a Banach space) converges weakly to $u$ iff $\{u_n\}$ is bounded and $u_n(t)$ converges weakly to $u(t)$ for each $t\in [a,b]$. Does this hold if we ...
2
votes
1answer
53 views

Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
0
votes
1answer
66 views

A linear bijection to a Banach space must have bounded inverse

Suppose that $X$ and $Y$ are Banach spaces, and $D ⊂ X$ is a linear subspace, which may not be closed. Suppose that $T : D → Y$ has a closed graph (in $X\times Y$), and is $1-1$ and onto. If $D$ is ...
2
votes
1answer
104 views

How can I visualize the nuclear norm ball

I want to see what the unit nuclear norm ball looks like. So I think of matrices whose singular values add up to $1$. For simplicity, let's talk about symmetric, $2\times 2$ matrices (so that I can ...
0
votes
1answer
27 views

For a given Hilbert space find a tight frame with bound A

For a given Hilbert space and $A>0$ find a tight frame with bound A. I know that an ortho-basis is a tight frame with $A=1$. Can I extend this to any $A>0$ by just scaling the ortho-basis?
1
vote
2answers
47 views

Weak* boundedness and norm boundedness in the dual of a normed vector space

Let $X$ be a normed vector space over $\mathbb R$, not necessarily Banach. Let $X'$ denote the dual of $X$, that is, the set of all bounded, linear functionals on $X$: $$X'\equiv\{f:X\to\mathbb ...
5
votes
0answers
54 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
10
votes
2answers
167 views

Can you equip every vector space with a Hilbert space structure?

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
1
vote
1answer
25 views

An Application: The Stone-Cech Compactification [duplicate]

If $X$ is completely regular, show that $X$ is open in $βΧ$ if and only if $X$ is locally compact.