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

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
0
votes
1answer
26 views

Spectral Measures: Pushforward

This thread is Q&A. Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
1
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1answer
28 views

Continuity of multiplication of operators in the strong operator topology - find an error

I need help in finding the mistake in the following reasoning. I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not ...
1
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1answer
20 views

Bound on the product of functions in $L^1$

Let X be a bounded subset of $\mathbb{R}$ and let $f, g,$ and $h$ be real valued functions in $L^2(X)$. Consider $$\| fgh\|_{L^1(X)}.$$ The hope is to get an upper bound in terms of ...
2
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2answers
38 views

How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
2
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0answers
33 views

A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq ...
1
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1answer
25 views

Evaluation function is Lipschitz wrt uniform conv metric

In the book on Brownian motion by Schilling and Praetzsch there is following statement: Let $\mathcal{C}_{(0)}:=\{f\in\mathcal{C}[0,\infty):\ f(0)=0\}$ be the space of all continuous functions ...
2
votes
0answers
41 views

The space of arrival for Fourier transform.

If $f\in L^2[-\pi,\pi]$, let $\hat f$ be the Fourier transform of $f$ $$\hat f=\frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ixt} dt, \ \ (-\infty<x<\infty)$$ we can see Fourier transform as an ...
2
votes
1answer
40 views

Normal Operators: Von-Neumann

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}N\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Regard their algebra: ...
3
votes
1answer
32 views

Name for a nonlinear version of bilinear form

A map $b:X \times Y \to \mathbb{R}$ is called a bilinear form if $b$ is linear in both arguments. Is there a name for a form $b$ which is linear in only one argument and may be nonlinear in the ...
3
votes
1answer
27 views

a counterexample in Muckenhoupt $A_1$ class.

We say $w$: $\mathbb R^N\to[0,+\infty)$ belons to Muckenhoupt $A_1$ class if there is a constant $C$ such that $$ \frac{1}{|B|}\int_{B(y)}w(x)dx\leq Cw(y) $$ for all $y\in \mathbb R^N$ and all balls ...
0
votes
1answer
138 views

The dual of $\mathcal{L}^\infty$

Let $X$ be the measurable functions with finite supremum norm on the unit interval. This is a banach space with respect to the supremum norm. The continuous functions on the unit interval have as dual ...
0
votes
1answer
21 views

Normal Operators: Examples

Given the Hilbert space $\mathbb{C}^2$. Consider bounded opertors: $$N:\mathbb{C}^2\to\mathbb{C}^2:\quad\|N\|<\infty$$ Then there are some with: $$N\neq N^*\quad N^*N=NN^*$$ What examples are ...
3
votes
1answer
78 views

Integrable functions that take values in a Banach space

Let $\mathbb K$ be $\mathbb R$ or $\mathbb C$. Let $(X, \mathcal M, \mu)$ be a measure space and let $F$ be a Banach space over $\mathbb K$. I would like to define an integral of a function $f:X ...
2
votes
1answer
25 views

How does this follow from the theorem?[normed linear space]

I have this theorem: Let X and Y be normed linear spaces and let $T:X\rightarrow Y$ be a linear transformation. The following are equivalent: a. T is uniformly continuous. b. T is ...
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0answers
67 views

Proof for Simplifying Integral involving Gaussian and Error Function

How do we simplify this integral? \begin{eqnarray*} \int_{-\infty}^{\infty}\left\{ ...
1
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1answer
77 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
-1
votes
1answer
33 views

Unboundedness of differential operator on test function

Currently I am studying the differential operator $T: L^2((0,1)) \to L^2((0,1))$ with the domain $D(T) = C_0^\infty((0,1))$. I am having difficulties finding a sequence to show the unboundedness of ...
1
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0answers
37 views

Special case for Riesz Representation Theorem

The usual Riesz Representation theorem can be found at page 49 in this book. Here I modify it a little bit and providing the following version Riesz representation theorem, where I replaced $\mathbb ...
1
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0answers
19 views

Multiplication by a Cutoff and Convergence in $H^s(\mathbb R^n)$

I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason: Let $K_j \subset ...
6
votes
3answers
96 views

Integration in Banach spaces

Let $X$ be a Banach space and let $L = \{f:[0,1]\to X\vert\, f \text{ Borel-measurable}, \int_0^1 \Vert f \Vert < + \infty \}$ ($\int$ being the Lebesgue integral.) Now define $$ T:L \to X^{**} ...
3
votes
2answers
57 views

What is the derivative of the inner product norm on $L^2$ space?

Let $f \in L^2(X)$ such that $f$ is generated by some arbitrary constant; that is, $f = g(a)$ with $g: \mathbb{R} \to L^2(X)$. Then what can be said about the derivative with respect to some arbitrary ...
0
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0answers
18 views

Signs conserved by positive operators

let $P_1$ and $P_2$ be two positive, linear and bounded operators on $L_2(0,1)$. For a given function $w \in C(0,1)$ if we define $y_1(x)=(P_1^{-1}w)(x) \quad \text{ and } \quad ...
2
votes
1answer
23 views

Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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0answers
23 views

Approximating the constant term of a polynomial

Let $P(T)$ be a polynomial in $T\in \mathbb R^n$ such that when $T$ is large, there is a function $F(T)$ such that $$|F(T)-P(T)|\leq Ke^{-\epsilon||T||}$$ where $K$ is a constant independent of $T$ ...
0
votes
1answer
25 views

Sobolev spaces, extensions and embeddings

I have the following statement whith an argumentation which I do not understand. Fix integers $k,l$ such that $0\leq l\leq k$. Then the identity map on $C^\infty(\mathbb{T}^d)$ extends to the ...
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0answers
28 views

Well-posedness of nonlinear PDE system

The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), ...
2
votes
0answers
46 views

Density of subset with nonlocal boundary condition

I am having difficulty proving that $E=\bigcap_{n\geq 0} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a dense subset of: $F=\{f\in C^2 (\mathbb{R}) : ...
0
votes
0answers
29 views

Behaviour of $L^2$ functions at infinity [duplicate]

Is it possible to prove that if $f\in L^2(\mathbb {R}) $ then $\exists\lim_{x\to\pm\infty}\lvert f\rvert^2$ and $\lim_{x\to\pm\infty}\lvert f\rvert^2=0$? If not, is it easy to find a counterexample?
2
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1answer
63 views

If $u_{n}\rightharpoonup u$ in $W_{0}^{1,p}\left(\Omega\right)$ , do we have $u_{n}^{+}\rightharpoonup u^{+}$? [closed]

If $u_{n}\rightharpoonup u$ in $W_{0}^{1,p}\left(\Omega\right)$ , do we have $u_{n}^{+}\rightharpoonup u^{+}$ and $u_{n}^{-}\rightharpoonup u^{-}$ in $W_{0}^{1,p}\left(\Omega\right)$ and vise ...
2
votes
0answers
41 views

Extensions of $C^k$ functions to the boundary [closed]

Assume $\Omega \subset \mathbb R{^n} $ is an open connected smooth domain. I have some propositions that I guess they are correct , but I want to be confident. If $f\in C_0^k(\Omega) $ then $f\in ...
3
votes
1answer
41 views

The derivative of a measure

Let $\mu$, $\nu$ be two Radon Measure on $\mathbb{R}^n$. How can I prove that $D_{\mu}{\nu}=\lim_{r \to 0} \frac{\nu(B(x,r)}{\mu(B(x,r))}$ is in $L^1_{loc}(\mathbb{R}^n,\mu)$?
2
votes
2answers
55 views

Need example of: Algebraic sum of closed vector subspaces need not be closed

I've read somewhere that given two closed subspaces $V_1,V_2$ in topological vector space $X$, their algebraic span $V_1+V_2=\{x_1+x_2 |x_i \in V_i, i=1,2\}$ need not be closed. I always thought that ...
0
votes
1answer
37 views

Closed convex hull of unitaries

If a C*-algebra ${\cal U}$ contains a non-unitary isometry $S$, show that $$\|S-A\|>\frac{1}{2n}$$ for every $A=\sum_{i=1}^n \lambda_iU_i$ which is the convex combination of $n$ unitaries. Thanks ...
1
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1answer
38 views

What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
1
vote
1answer
45 views

proof of existence of a solution with $ f \in L^1$

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in L^1(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ for the problem ...
1
vote
1answer
41 views

Existence of unbounded operators on Banach spaces

I'm confused by the questions Discontinuous linear functional and Example of an unbounded operator which ask about unbounded linear functionals/operators on Banach spaces. I don't understand how ...
0
votes
2answers
60 views

Show linear map is continuous

Consider the linear map $T : \mathbb{R}^3 \to \mathbb{R}^3$ as: $T(x_1, x_2, x_3) = (x_1+x_3, x_2-x_1, x_3)$ I know that every linear map is contiuous if the vector space $X$ is finite dimensional. ...
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vote
2answers
64 views

Can someone give me a hint on how to solve this question

Let $(X,d)$ be a compact metric space. For each $n \in \mathbb{N}$ we have $ f_n:X \to \mathbb{R}$ be a continuous function such that $f_n(x) \geq0 \forall x \in X$ . Assume that for all $x \in X$ the ...
0
votes
1answer
39 views

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
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votes
1answer
58 views

Is it incorrect to say that a functional “maps functions to numbers”?

Does a "functional" always takes in a function and spit out a number? This is what a professor said in class a long time ago but now I am studying Frechet derivative and a claim was made that a ...
2
votes
1answer
55 views

Continuity at $x=0$ of this function

Not a hard exercise:$$f(x)=\frac{1}{x^3}\cdot \int_{-x}^x \sin(4t^2) \, \text{d}t \quad \text{where} \space x\ne 0\:$$ $$f(x)=5\:;\:x=0\:$$ Checking it's continuity at $x=0$ by using L'Hospital's ...
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0answers
24 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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votes
1answer
52 views

Domain Issue: Notation

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ It is well known that:* $$A=A^{**}\iff ...
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0answers
43 views

Uniform convergence of functions involving normal CDF

Consider two sequences of continuous functions $(f_n)$ and $(g_n)$ for $n \geq 0$ defined by $$ f_n (x) := \int_0 ^t \Phi\left(\frac{x\Phi ^{-1}(\alpha(s) + \beta_n(s))+\Phi^{-1} ...
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0answers
19 views

$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
0
votes
0answers
15 views

Fredholm determinant for general kernel (discontinuous function, distribution, etc.)

For a Fredholm equation of the second kind $\phi(x) - \lambda \int_a^b k(x,y) \phi(y) dy = f(x)$, the solution can be obtained by constructing a solving kernel for any regular value of $\lambda$, for ...
1
vote
1answer
29 views

Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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votes
2answers
39 views

Prove Weierstrass function has a pole of order 2 for all $\omega \in L$

I need some help to prove Weierstrass function has a pole of order 2. The Weierstrass function $\wp$-function of lattice $L$ is defined by $$\wp(z) = \wp(z; L) = \frac{1}{z^2} + \sum_{w \in ...
0
votes
1answer
43 views

Is linear dual space a misleading term?

A linear dual space consists of all linear functionals that sends a function in the space $X$ to its underlying field But the linear space itself does not send element of the field to the space $X$, ...