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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3
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2answers
59 views

A Banach Space cannot have a denumerable basis:Why is it true?

I came across the following theorem: A Banach Space cannot have a denumerable basis which has been proven in my book. I can't understand why is it true since $\mathbb R$ is a banach space over ...
2
votes
1answer
24 views

About closedness and boundedness of $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$

Let $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$. To check which one is true: (a) $H$ is bounded (b) $H$ is closed (c) $H$ is a subspace (d) $H$ has interior points My ...
0
votes
1answer
27 views

Infinite Hamel basis for Banach spaces

What are some standard examples of Hamel basis for Banach spaces with cardinality >= $\aleph_0$? I tried searching, but couldn't find any.
0
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0answers
37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...
0
votes
1answer
36 views

space of all lipschitz maps is a polish metric space

Suppose that $(X, d_X)$ and $(Y, d_Y )$ are Polish metric spaces. Let $L(X, Y )$ denote the set of all Lipschitz maps from $X$ to $Y$ with the pointwise convergence topology. Show that $L(X, Y )$ is ...
1
vote
0answers
25 views

Application of Riesz-Markov-Kakutani representation

Let A be finite set, $\{f_i\}_{i=1}^{n}:A\to \mathbb{R}$ non-negative and with the following property for every $\sum \lambda_{i}=1$: for any $g=\sum \lambda_{i}f_i$ there exists $a\in A$ s.t. ...
2
votes
2answers
52 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
0
votes
1answer
37 views

A closed subspace of a separable Hilbert Space is Separable

Suppose $X$ is a Hilbert Space which is separable. Let $Y$ be a closed Subspace of $X$. I need to show that $Y$ is separable. Since $X$ is separable it has a countable dense subset say $M$. Taking ...
0
votes
0answers
27 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
3
votes
1answer
21 views

Are all normed linear spaces Hausdorff? What about a bounded subset of a normed linear space?

The proof that the dual space of a normed linear space is complete in proposition 5.4 of chapter on Banach spaces (John Conway, functional analysis) consists of restricting the functionals in the dual ...
2
votes
0answers
28 views

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
votes
0answers
24 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
votes
0answers
27 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
votes
0answers
63 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
votes
0answers
47 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 ...
-2
votes
1answer
62 views

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

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 $
1
vote
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 ...
0
votes
0answers
43 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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
23 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
vote
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 ...
1
vote
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 ...
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
25 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 ...
1
vote
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
vote
1answer
35 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
63 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, ...
-2
votes
1answer
46 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 $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
1
vote
1answer
28 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 ...
0
votes
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": ...
0
votes
0answers
25 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
votes
0answers
40 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
34 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
42 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
26 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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votes
0answers
24 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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vote
0answers
33 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
47 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
vote
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
votes
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
70 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
34 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
128 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 ...