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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94 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
1
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3answers
63 views

Size of function spaces

For example, how big is the space $ C^k(\mathbb{R},\mathbb{R}) $ ? How much is, say, $ C^0 $ larger than $ C^1$ ? How can one figure out ?
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0answers
57 views

Laplace Transform of $e^{a t^2}$

What is the Laplace transform of $e^{a t^2}$, for positive $a$? In order for Laplace transform to exist function must be locally integrable. Since integral of any compact set $e^{a t^2}$ is finite ...
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1answer
46 views

Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$

Is Fourier transform method suitable for to solve the following equation \begin{align*} \int g(x-t)e^{-t^2/2} dt = e^{-a|x|} \end{align*} Suppose we take the Fourier transform of the above ...
3
votes
0answers
90 views

Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
3
votes
1answer
155 views

Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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2answers
44 views

Proving compactness of an operator

I'm having a hard time finding a solution for the following problem: Prove that the operator $ T \in \mathcal{L}(\ell_2) $ defined with the formula $$ T((x_1, x_2, \dots, )) = (0, x_1, x_2/2, x_3/3, \...
0
votes
1answer
168 views

Complement of the Image is the Image of the Complement

Given a continuous linear map $T:E\to F$ where $E,F$ are normed vector spaces, I am wondering about the trivial question whether for any subset $U\subset E$, it holds or not: $$T(U^c)=T(U)^c$$ I could ...
0
votes
1answer
145 views

Strict Bessel inequality in $\mathcal{l}^2$

I'm asked to give an example of an $x\in \mathcal{l}^2$ s.t. $$\sum_{j=1}^{\infty}|<e_j,x>|^2<||x||^2$$ Where $(e_j)$ is some orthonormal sequence. However, I think this question is more ...
0
votes
1answer
47 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
1
vote
1answer
366 views

Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
4
votes
1answer
666 views

On a proof of Riesz-Fischer Theorem

Questions : [See below for context.] $\rm\color{#c00}{a)}$ First, is the proof presented below $100$ % correct ? $\rm\color{#c00}{b)}$ How would one justify the LHS of $(2)$ ? Are my ...
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1answer
62 views

is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$?

Consider the $L^p$ spaces. is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$? is it true if the domain of $L^p$ is finite measure? Thanks
2
votes
0answers
53 views

Coercivity of a bilinear form

Consider a very smooth open and bounded set $\Omega$ of $\mathbb{R}^d$. One can prove that for all $\epsilon >0$, $\exists K_{\epsilon}>0$, such that for all $u \in H^1(\Omega)$ : $\|u_{/\Gamma}...
1
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2answers
158 views

Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
4
votes
3answers
209 views

condition for equivalence of norms on vector spaces

Let us call two norms $|x|_1$ and $|x|_2$on a finite-dimensional vector space equivalent if they set the same topology on that space. I need to show that this definition is equivalent to the existence ...
3
votes
1answer
69 views

Is $x\mapsto \| Tx\|$ lower semi-continuous?

Suppose $T:\mathcal D(T)\rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$. Is it true that $$ \|Tx\|\leq \liminf_{n\rightarrow\infty} \|T x_n\...
1
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2answers
52 views

Norm of Functional on : $c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$

Take $E = c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$ and define on $E$ the functional: $$F(x) = \sum_{n=1}^\infty \frac{x_n}{2^n}$$ $\cdot$Show that $F$ is a linear continuos functional on $...
2
votes
0answers
83 views

Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$\max~f(x)\quad \mbox{s.t.}~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can ...
0
votes
1answer
81 views

Norm of Integral Operator on $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$

There are similar question but the characterization of the space $E$ that I have gives me problem in computing the actual norm. Let $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$ with the usual $\parallel \cdot\...
11
votes
2answers
224 views

General “Hodge theorem”

I know basically zero Hodge theory, so this question might be weird. Let $$A \stackrel{S}{\longrightarrow} B \stackrel{T}{\longrightarrow} C$$ be a sequence of closed, densely defined maps of ...
2
votes
1answer
73 views

Is the Riemannian distance function Lipschitz on a hypersurface?

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$. Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian ...
1
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1answer
23 views

Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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votes
3answers
83 views

Why is $R-\lambda$ invertible for $|\lambda|<1$

I got the following question: Why is $R-\lambda$ invertible for $|\lambda|>1$ and not invertible for $|\lambda|\leq1$ ? R is the right shift operator on $\mathfrak{l^2}$
1
vote
1answer
44 views

Prove subset of $\mathbb{C}^n$ is convex and complete

I have to prove that the subset $M=\sum_{i=1}^n x_i=1$ of $\mathbb{C}^n$ is convex and complete w.r.t. the inner product $<x,y>=\sum_{i=1}^n x_i\bar{y_i}$. Now being convex is trivial. However ...
1
vote
1answer
114 views

weak* convergence definition in Sobolev space

I have a question which might quite trivial but I would appreciate any assistance. Why does it follow that for Sobolev spaces, say $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$, it follows ...
3
votes
1answer
325 views

Dual space of L infty space

The Dual space of L infty space is not L1 ,are there some example to show this? I am going to use rieze representation thm,but it can not be used because p= $\infty$.
0
votes
1answer
42 views

Geometrical meaning of a face

Let $(X,P)$ be a locally convex space, $K$ a compact, convex subset of $X$. A face $F$ of K is a nonempty, compact, convex subset of $K$ s.t. $$\forall y,z\in K \,\forall t\in(0,1) \left[ (1-t)y + tz \...
0
votes
1answer
197 views

The uniform convergence of the difference quotient of a smooth compactly supported function

This question is the same one as the question found here: Explanation on a proof of a property of mollifiers However, the answers given just confuse me more. The question asked in that thread is ...
0
votes
1answer
139 views

Bounded Right Inverse

If a linear operator between two Banach spaces is surjective and bounded, can we get any information about a right inverse? For example, is it bounded? Thanks, trying to understand trace operator ...
7
votes
2answers
554 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
87 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
111 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
66 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
50 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
117 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
36 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
51 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 L^2(\...
1
vote
2answers
52 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 S\varphi,\psi\rangle=\langle\varphi,T\psi\...
0
votes
1answer
61 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
57 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 }~ W^{...
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votes
0answers
179 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: ...
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0answers
57 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 ...
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0answers
47 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 $\mathbb{R}$-...
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2answers
191 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
188 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
7
votes
1answer
171 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
189 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
25 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 $0<L&...
1
vote
0answers
215 views

Mollifier proof, Evans PDE [duplicate]

My question is related to the mollifier properties in the appendix of Evans' PDE. In proving that $f^{\varepsilon}$ is smooth, he constructs the difference quotient, with the original integral over $...