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, ...

learn more… | top users | synonyms (1)

2
votes
0answers
46 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
0answers
51 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}) : ...
2
votes
0answers
48 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 ...
2
votes
1answer
57 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 ...
2
votes
0answers
46 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} ...
2
votes
0answers
105 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
2
votes
1answer
75 views

Existence of specific weak derivative

Suppose there exits a sequence $(\phi_n)_n\subset C_0^\infty(\Omega)$, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with $C^\infty$ boundary, such that $(\partial_1+\partial_2)\phi_n$ ...
2
votes
1answer
87 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
2
votes
0answers
76 views

Is the Laplacian an unbounded operator?

"The Laplacian is an unbounded operator": I read this in a book. But on Wikipedia it says: The Laplace operator $$\Delta:H^2({\mathbb R}^n)\to L^2({\mathbb R}^n) \,$$ (its domain is a Sobolev ...
2
votes
0answers
59 views

$\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$

I have a encountered to a problem in reading article . Can someone look at the page 9 in this article and give a hint that why $\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < ...
2
votes
1answer
91 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in ...
2
votes
0answers
60 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...
2
votes
0answers
32 views

Is $L_{P}(R)$ Banach algebra with multiplication defined by convolution?

Is $L_{P}(R)$ with convolution a Banach Algebra?
2
votes
1answer
52 views

Lifting idempotents from a quotient of a Banach algebra

In a quotient of a Banach algebra $A$, if an invertible element is connected to the identity by a continuous path of invertibles, then it can be lifted to an invertible element in $A$. Is there an ...
2
votes
1answer
56 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
2
votes
1answer
37 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
2
votes
1answer
78 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
2
votes
2answers
28 views

Finding adjoint of an operator from $\mathbb{C}^n$ to $H$

Suppose we have vectors $h_1,\ldots,h_n \in H$, where $H$ is a Hilbert space. Define $B : \mathbb{C}^n \to H$ by $$B(z_1,\ldots,z_n)=\sum_{j=1}^n z_j h_j.$$ Calculate $B^* : H \to \mathbb{C}^n$. ...
2
votes
1answer
59 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
2
votes
0answers
31 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
2
votes
1answer
79 views

Linear Operators Denseness and Injectivity

I'm studying for a Real Analysis prelim and have the following problem: "Let $X$ and $Y$ be normed spaces over $\mathbb{R}$ and let $$\mathcal{L}(X, Y) = \{T: X \rightarrow Y \mid T \text{ is bounded ...
2
votes
0answers
51 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
2
votes
0answers
179 views

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
2
votes
0answers
79 views

If one expresses a function as a linear combination of other functions, can the linear combination relationship be inverted?

If one has a function say f, that can be expressed as a linear combination of another type of function say g, can one invert the relationship as a linear combination of the other function? i.e. if one ...
2
votes
0answers
81 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
2
votes
1answer
48 views

What is a integral of operators?

I am reading the book Semigroups of Linear Operatos and Applications to Partial Differential Equations which studies a uniformly continuous semigroup, this is a family $(T_t)_{t \geq 0}$ of bounded ...
2
votes
0answers
55 views

Span of Polynomials in $\mathcal{C}(\mathbb{R})$ [duplicate]

Let $\mathcal{P}=\{1, x, x^2, x^3 \ldots\}$. Then pick out the correct statements. A) Span$\mathcal{P}=\mathcal{C}(\mathbb{R})$ B) Span$\mathcal{P}$ is a subspace of $\mathcal{C}(\mathbb{R})$ C) ...
2
votes
0answers
45 views

Dimension of a Hilbert space

Halmos in his book (A Hilbert space problem book) says, 1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. 2- Also he proves if orthogonal dimension of ...
2
votes
1answer
42 views

A question in Hahn-Banach theorem

Let $X$ is a real vector space(without topology). call a point $x \in A \subset X$ an internal point of $A$ if $A-x$ is an absorbing set.Suppose $A$ and $B$ are disjoint convex set in $X$ and $A$ has ...
2
votes
0answers
75 views

Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
votes
0answers
83 views

Inner measure (inner set function) on functional closed sets

I'm struggling with the following problem: Let $X$ be a set and $\mathcal{Z}:=\{Z\subseteq X \,\big|\,\exists\,\psi\in\mathcal{C}(X)\,:\,Z=\psi^{-1}(\{0\})\}$ the family of functional closed sets ...
2
votes
0answers
22 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
2
votes
0answers
20 views

Find the inverse of an operator, and determine is it bounded.

I've been doing some similar problems, but I got stuck on this one... and I have a feeling I'm running in circles trying to solve it. Any help appreciated! Problem: We have an operator: $$ A : ...
2
votes
1answer
35 views

Adjoint of canonical expansion of compact operator

Lets say I have given a rank-$n$ operator $A = \sum^n_{k=1} \lambda_k \langle u_k, \cdot \rangle v_k$. Then it is straightforward to compute its adjoint as $A^\ast = \sum^n_{k=1} \lambda_k \langle ...
2
votes
0answers
40 views

Compatibility of operator spaces and tensor product norm

I have a problem with understanding the notion of complete boundedness in tensor notation. One possible way of saying a mapping $\phi\colon \mathcal{A}\to \mathcal{B}$ is completely bounded is to ...
2
votes
2answers
56 views

$||f||_1 =(\int_a^b [|f|^2+|f'|^2]dx)^{1/2}$. Is this normed space complete?

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$||f||_1 =\left(\int_a^b \left(|f|^2+|f'|^2\right) dx \right)^{1/2}$$ Is this normed space ...
2
votes
0answers
64 views

Hahn-Banach theorem application

I don't understand how by Hahn Banach theorem we have the following: Suppose that there exists $f_1, f_2,..$ holomorphic in a Riemann surface $R$ which is zeros on a sequence of points $x_k$ and non ...
2
votes
1answer
39 views

Functional differentiation involving the derivative of the function.

I've recently come across this functional: $F[f] = \int \frac{|\nabla f|^2}{f} dr$ (in 3D space, if that's relevant) and am interested in taking the functional derivative $\frac{\delta ...
2
votes
0answers
41 views

Extension of Riesz Representation Theorem for locally compact Hausdorff space.

Riesz representation theorem for positive linear functional is well known when underlying space is locally compact Hausdorff space. In this case, the measure we obtain is a positive measure. For ...
2
votes
1answer
47 views

Riesz representation theorem

Suppose $\Lambda$ is a bounded linear function on a Hilbert space $H$, given by an inner product with a unique fixed vector $h_0 \in H$ such that $\Lambda(h) = \langle h,h_0 \rangle$. Set $M = \ker ...
2
votes
1answer
107 views

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$.

$M$ and $N$ are subspaces of a Hilbert space. If $M\subset N$, show that $N^{\perp}\subset M^{\perp}$. Show also that $(M^{\perp})^{\perp}=M$. I know that the orthogonal complement of $X$ is the set ...
2
votes
0answers
43 views

Infinite dimensional space with metric generated by norm

In my classwork I have to provide examples of infinite dimensional vector spaces with metric generated by their norm. I readily provide space of continuous function $C[a, b]$ and Hilbert sequence ...
2
votes
0answers
93 views

A particular decomposition of a CPTP map

Let $\mathfrak{D}$ denote the set of $n\times n$, trace-one, positive semi-definite matrices (known as density matrices in quantum information theory). Consider a Completely Positive Trace Preserving ...
2
votes
0answers
44 views

Characterization of Bochner dual

I want to prove following theorem Let X be separable and reflexive Banach space, $1<p<\infty$ than $$ L^p((0,1),X)^* = L^q((0,1),X^*) $$ where $\frac1{p}+\frac1{q} = 1$, with ...
2
votes
1answer
35 views

The functional is continuous

Show that the functional $J(y)=\int_a^b (\sin^3 x+y^2) dx$ is continuous in respect to the $||\cdot||_{\infty}$ norm, at any $y_0 \in C([a,b])$. Let $y_0 \in C([a,b])$. Then for $y \in C([a,b])$ we ...
2
votes
0answers
91 views

Request for a comparison between these 3 (advanced?) functional analysis books?

It would be helpful if I can get some comparison between these three books, T. Tao, An epsilon of room, I, Graduate Series in Mathematics 117, American Mathematical Society (2010). T. Tao Analysis ...
2
votes
1answer
51 views

In the space of probability distributions, is the set of discrete distributions dense?

Is the following true: In the space of probability distributions, the set of discrete distributions is dense with regard to the Lévy metric. Can someone point me to any reference on this ...
2
votes
1answer
44 views

Integrating a linear-map valued function

My textbook for an Analysis course I am taking presents the Mean Value Equality theorem as Suppose $\mathbb{X}, \mathbb{Y}$ are Banach Spaces. Let $U\subseteq \mathbb{X}$ be an open set, and let ...
2
votes
1answer
48 views

Subsec. 4.1-8 in Kreyszig's functional analysis book: Does every inner product have a total orthonormal set?

In every Hilbert space $H \neq \{0 \}$, there exists a total orthonormal set. I think I've understood the proof given by Erwin Kreyszig in Introductory Functional Analysis With Applications. ...
2
votes
0answers
58 views

The functional is not continuous in respect to the strong norm

Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1:= ||y||_{\infty}+||y'||_{\infty}, y \in V$. But the converse is ...