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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For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
2
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0answers
20 views

Orthogonal Complement: Families

Problem Given a Hilbert space $\mathcal{H}$. Consider a family: $$A:\Lambda\to\mathcal{P}(\mathcal{H}):\lambda\mapsto A_\lambda$$ Remind that: $$A\subseteq\mathcal{H}:\quad ...
0
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0answers
26 views

The convergence of the norm implies the convergence of the sequence via the norm. [duplicate]

I want to prove that: Let $(f_n)$ be a sequence in $L^p$ that converges pointwisely to $f$. Prove that $f_n\rightarrow f$ in $L^p$ iff $||f_n||\rightarrow ||f||$. The "only if" is easy and follows ...
2
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62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
2
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40 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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1answer
62 views

this series converges and how can I prove it? [on hold]

I need to prove that this series converges . Note that the series is indexed all integers $ \sum_{k\in\mathbb{Z}}\vert\lambda\vert ^{k}, \; \; \; with \; \vert\lambda\vert<1 $
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1answer
39 views

Definition in Operator Theory

I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it ...
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0answers
16 views
+50

Kernel, Green function and the functional derivative.

I am pretty new to the subject of differential equations and am reading about Green functions and Kernels for the first time. I am more familiar with functional differentiation and am comfortable with ...
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0answers
26 views

Meaning of Finite-dimensional subspace of $C_0(\Omega,\mathbb R^N)$.

I am reading a paper which has something to do with a finite-dimensional subspace of $C_0(\Omega,\mathbb R^N)$. In this paper, it about to compute a value defined as $$ \sum_{n=1}^N \alpha_j\| A_j ...
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19 views

approximate unit of $K(H)$- ordering on $K(H)$ and finite rank operators

Let $H$ be a complex Hilbert space with orthonormal basis $\{e_i:i\in I\}$ . Consider the $C^\ast$-algebra of the compact operators on $H$, $K(H)$. For a finite subset $F\subseteq I$, let $P_F$ be the ...
1
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1answer
27 views

A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Functional Analysis.

I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question 1.5a, he gives as a reference for this question the book by Wilkinson called "The Algebraic Eigenvalue ...
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2answers
26 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space ...
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27 views

$A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\} \Rightarrow$A is closed [closed]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
1
vote
1answer
21 views

To show that $y$ is the best approximation of $x$ from $G$ i.e $y$ is the unique element of $G$ such that $||x-y||=d(x,G)$

Let $G$ be a closed subspace of a Hilbert Space $H$. For $x \in H$, let $y$ be the orthogonal projection of $x$ on $G$. Then I need to show that $y$ is the best approximation of $x$ from $G$ i.e $y$ ...
0
votes
1answer
24 views

Series of positive-definite kernels

Suppose I have a positive definite, shift invariant kernel $k_1(x-y)=k_1(\delta)$. I want to know whether the sum (where $a_n\geq 0$) $$ k(\delta) = \sum_{n=1}^{\infty} a_n k_1(n\delta)\tag{*} $$ is ...
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0answers
14 views

What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
1
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1answer
34 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
4
votes
2answers
62 views

The density of polynomials in the space of continuous functions on the unit ball of $\ell^p$

Let $$B = \{a : \|a\|_p \le 1\} \subset \ell^p(\mathbb{N})$$be the unit ball, endowed with the weak topology. For which $p$, where $1 < p \le \infty$, are the functions of the form$$f(a) = q(a_0, ...
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1answer
41 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1*X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff topological ...
1
vote
1answer
24 views

Continuous function not sobolev

Let $I=(a,b)$ an open bounded interval. It is well known that $W^{1,p}(I)\subset C(I)$. It easy to see that there are $f\in C(I)$ such that $f\notin W^{1,p}(I)$ It is enough to take $I=(0,1)$ and ...
3
votes
1answer
58 views

Example of a function in $L^2(\mathbb{R})$ with derivative not in $L^2(\mathbb{R})$.

We know examples of a function which doesn't lie in $L^2(\mathbb{R})$ with derivatives in $L^2(\mathbb{R})$: $$f_1(x) = \mathrm{arctg}(x) \notin L^2(\mathbb{R}), \qquad f_1'(x) = \frac{1}{x^2+1}\in ...
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0answers
15 views

For what does the formula $(\prod_{t=1}^d[\begin{array}{c}-\frac 12&1&-\frac 12\end{array}]_{l_t,i_t})f$ stand for?

Let $f:\mathbb R\to\mathbb R$ and $$a_{l,i}:=f(x_{l,i})-\frac{f(x_{l-1,(i-1)/2}+f(x_{l-1,(i+1)/2})}2$$ for some $x_{l,i}$. I've read, that we can write $a_{l,i}$ in the following "operator form": ...
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0answers
23 views

Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

Let $X$ be a Hilbert Space and let $\{S\}$ be a Convex set in $X$. Let $d=\inf_{x \in S}\|x\|$ . Prove that, if $\{x_n\}$ is a sequence of elements in $S$ such that $\lim_n \|x_n\|=d$, then $\{x_n\}$ ...
3
votes
1answer
52 views

$S = \left\{ x^* Ax\mid x \in C^n ,\ x^*x = 1 \right\} \implies S\;$ is compact and convex

Let $\,A \in {\mathbb{C}^{n \times n}}\,$ and $\,S = \left\{ {{x^*}Ax \mid x \in \mathbb C^n,\ {x^*}x = 1} \right\}.\,$ Why is $A$ compact and convex?
2
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36 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
0
votes
0answers
10 views

Spectrum of hypercyclic operators

Can we say the boundary spectrum of (non-quasinolpotent) hypercyclic operators is not included in point spectrum?
-1
votes
1answer
27 views

Upper bound on a integral

Let $h(x)$ be a smooth periodic function on $(0,T)$. $\int_0^Th(x)\,\mathrm{d}x=c\in(0,\infty),h(x)\in(0,1)\,\forall\, x\in(0,T)$ Possible to obtain an upper bound of ...
1
vote
2answers
33 views

infinite dimensional hilbert space - uniqueness of series expansion

A function $f(x)$ is expanded in a series of orthonormal functions $$ f(x) = \sum_{n=0}^{\infty} a_n \varphi_n(x) $$ Show that the series expansion is unique for a given set of $\varphi_n(x) $. The ...
1
vote
0answers
41 views

Can we have different methods to estimate elements from Lp spaces?

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the ...
1
vote
1answer
24 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
0
votes
1answer
32 views

An isomorphism between two Banach algebras

Consider the compact set $[-1,1]$ and $C([-1,1])$ the set of all continuous functions $\phi: [-1,1] \rightarrow \mathbb{C}$. I want to show that the quotient of $C([-1,1])$ by $\mathbb{C}$ is the ...
0
votes
1answer
26 views

Do convergence a.e. + limit function being in $L^p$ imply $L^p$ convergence?

Suppose $f_n\in L^p$ such that $f_n \to f$ almost everywhere. If we further know $f \in L^p$, can we say that $f_n \to f$ in $L^p$ norm?
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23 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
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0answers
30 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
3
votes
1answer
39 views

What does the symbol $H^1_0(\Omega)$ mean?

Here $\Omega \subset \mathbb{R}^n$ is a closed disc centred at $0$ with radius $r$.The book I am reading is assuming the Dirichlet boundary condition on $\Omega$ and claiming that the dual of ...
1
vote
1answer
46 views

$(T_n)_{n\in\mathbb{N}}\subseteq L(H)$, $T_n\to T$ weak, why does there exist $C>0$ such that $\|T_n\|\le C$ for all $n\in\mathbb{N}$?

Let $H$ be a Hilbert space, $(T_n)_{n\in\mathbb{N}}\subseteq L(H)$ a sequence such that $T_n^*=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a map $T\in L(H)$ such that $T^*=T$ ...
1
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1answer
31 views

Definition of space $L^2(\mu)$ where $\mu$ is a Borel probability measure on $\mathbb R$.

Let $\mu$ be a Borel probability measure on $\mathbb R$ with compact support. Consider the space $L^2(\mu)$. It is the first time that I meet this space (usually I have $L^2(\mathbb R)$). Is it still ...
2
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0answers
45 views

If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
3
votes
1answer
68 views

Prove the Lipschitz constant must be less than 1.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
2
votes
2answers
33 views

The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
3
votes
1answer
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
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0answers
21 views

Integral of Laplacian eigenfunctions square

The Laplacian densely defined in $L^2(\mathbb{R}^3)$ has eigenfunctions $f_k(x)$ that are defined as generalized functions. I need to define the integral of the square of these eigenfunctions in a ...
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1answer
24 views

In every infinite-dimensional TVS, every w-neighborhood of 0 contains an infinite-dimensional subspace (Rudin's FA, p. 66))

In Rudin's Functional Analysis, second edition, p. 66 I bumped into the following proposition: If X is infinte-dimensional [topological vector space with a dual that separates points on X] then every ...
2
votes
1answer
54 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
2
votes
1answer
33 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
4
votes
2answers
41 views

Given: self-adjoint, monotonic increasing sequence in $L(H)$ such that $\|T_n\|<C$. Why converges $(T_n)$ strongly to a self-adjoint $T\in L(H)$?

Let $H$ be a Hilbert space, $(T_n)\subseteq L(H)$ a sequence such that $T_n^\ast=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a constant $C>0$ such that $\|T_n\|<C$ for all ...
3
votes
1answer
25 views

Does $(\ell^{1}(\mathbb Z), \cdot)$ have a bounded approximate identity?

Put $\ell^{1}(\mathbb Z)=\{f:\mathbb Z \to \mathbb C: \|f\|_{\ell^{1}}:=\sum_{n\in \mathbb Z}|f(n)|< \infty \}$ and we note that $\ell^{1}(\mathbb Z)$ is an algebra under pointwise multiplication. ...
0
votes
1answer
25 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
0
votes
0answers
32 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
4
votes
0answers
62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...