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)

1
vote
1answer
32 views

Non-existence of a continous-norm on a sequence space.

For $U\cong \prod_{n\in \mathbb{N}} \mathbb{R}$ equipped with the product topology, i have already shown, that $U$ is a Frechet-Space w.r.t. the frechet-metric. How to prove that there exists no ...
1
vote
0answers
22 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
1
vote
0answers
25 views

Different definitions of Morrey and Campanato Spaces

The book by Giaquinta defines Campanato spaces using the seminorm: $$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ ...
1
vote
0answers
16 views

Existence of Galerkin approximations to PDE

Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
1
vote
0answers
22 views

functions with several variables: show injectivity

Let $a,b \in \mathbb R,~ 0 < a < b$. Let $f=(f_1,f_2,f_3): \mathbb R^2 \rightarrow \mathbb R^3$ be defined by: $f_1(s,t) = (b+a \cdot cos(s))cos(t)\\f_2(s,t)=(b+a \cdot ...
1
vote
0answers
21 views

Statement of Markov-Kakutani fixed-point theorem

Markov-Kakutani fixed-point theorem is usually stated as follows: "Let $E$ be a locally convex topological vector space. Let $C$ be a compact convex subset of $E$. Let $S$ be a commuting family of ...
1
vote
0answers
18 views

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ...
1
vote
1answer
59 views

The space of distributions endowed with the topology of uniform convergence on bounded sets is not Fréchet.

I found a state, that the space of distributions on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Fréchet space. As far as i can ...
1
vote
0answers
41 views

Find the adjoint operator.

Consider the sequence space $\ell_p$ and S defined by $(1\leq p<\infty)$$$ S:\ell_p\to\ell_p:(x_1,x_2,x_3,\ldots)\mapsto(0,x_1,x_2,\ldots) $$ Find the $S^*$ operator.
1
vote
1answer
33 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
1
vote
1answer
42 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
1
vote
0answers
28 views

Simple tensors in the dual space

Let $X$ and $Y$ be two Banach spaces and assume, if necessary, that $X^*, Y^*$ have the approximation property (but not necessarily the Radon–Nikodym property). Consider the injective tensor product ...
1
vote
1answer
86 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
1
vote
0answers
66 views

Zabreiko's Lemma

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p: X \to [0,\infty)$ be a seminorm. If for all absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ ...
1
vote
0answers
42 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
1
vote
1answer
26 views

seminorms and product topology

I'm studying Conway's functional analysis by myself. So some questions which may be simple is not clear for me. My question is the following: If $\{X_i: I\in I\}$ is a family of TVS, then $X ...
1
vote
1answer
33 views

weak$^∗$ neighborhood of $x$ in $\ell_1$

I have this problem Let $x \in \ell_1$ and $\epsilon>0.$ Choose an $N\in N$ such that $\sum\limits_{k=N}^{\infty}|x_k|<\epsilon$ I cannot understand why V is a weak$^∗$ neighborhood of $x$ in ...
1
vote
1answer
23 views

Balanced Core: Explicit Expression?

Denote the collection of all balanced subsets by: $\mathcal{B}:=\{B\subseteq X: B\text{ balanced}\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary ...
1
vote
1answer
36 views

Exists $l\in X^*$ such that $\|l\|=1, l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$?

Let $(X, \|.\|)$ be a normed space and $Y\subsetneq X$. How can you prove that for $x_0 \in X\setminus Y$ there exists $l\in X^*$ such that $\|l\|=1$, $l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$ ...
1
vote
1answer
33 views

How implies “$\sum_{k=1}^\infty z_nx_n$ exists for every x $\in c_0$” that z $\in l^1$?

If z is a sequence in $\mathbb R$ and $\sum_{k=1}^\infty z_nx_n$ exists for every x $\in c_0$, how follows that z must be in $l^1$?
1
vote
1answer
15 views

type I algebras definition

I'm studying von Neumann algebras type decomposition and I already noticed two "different" definitions of, for instance, type I : (i) a vN algebra is said to be of type I, if it has an abelian ...
1
vote
1answer
49 views

> Complemented subspace of E'''

Let $E$ normed space. Is $E'$ a complemented subspace of $E'''$? Can you help me?
1
vote
0answers
42 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
1
vote
0answers
18 views

Estimates in Hölder spaces

Let $u,v\in C^{2,\alpha}\left(\overline{\Omega}\right)$. Proof that there exists a constant $C>0$ so that \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq ...
1
vote
0answers
48 views

Cauchy-Schwarz type formula for positive integral operator

Let $\gamma(x,y)$ be some complex valued function in $L^{2}(\mathbb{R}^{2})$ such that $$ \gamma(x,y)=\overline{\gamma(y,x)},\forall x,y\in \mathbb{R} $$ Let $S=(1-\Delta)^{1/2}$ acting on $\gamma$. ...
1
vote
1answer
58 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, ...
1
vote
0answers
30 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
1
vote
1answer
37 views

inductive limit

Consider spaces $$E_n=\{x=\{x_k\}_{k\in\mathbb{N}}\mid x_j=0,\quad j>n\},\quad x_k\in\mathbb{R}$$ endowed with $\|\cdot\|_\infty$ norms. Let $E$ be an inductive limit of these spaces. This set ...
1
vote
0answers
27 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
1
vote
0answers
29 views

compare spectrum radius for two self adjoint, positive definite operators

I have two self adjoint operators $T_1$ and $T_2$, both of them are positive definite. And $T_1$>0, $T_2$>$T_1$. I'm wondering whether it's true that the spectrum radius of $T_2$ is greater than ...
1
vote
0answers
21 views

A second question about a proof of Banach-Alaoglu

I have another question about the proof of Banach-Alaoglu using nets. The proof proceeds by considering a universal net into the closed unit ball of $X^\ast$, let's call the ball $S$ and the net ...
1
vote
0answers
28 views

Equivalent Frechet differentiable norm on $\ell_1$ and $c_0$

Does there exist an equivalent Frechet differentiable norm on $\ell_1$ and $c_0$? I think we can not find an equivalent norm on $\ell_1$ but we could find an equivalent norm on $c_0$, I do not prove ...
1
vote
0answers
36 views

Cartesian product of reflexive spaces

Given $(E,\|\|_E),(F,\|\|_F)$ reflexive normed vector spaces. I have to prove that also $(E\times F,\|\|_{E\times F})$ is reflexive where $\|\|_{E\times F}$ is the product norm. What I know is that ...
1
vote
0answers
47 views

Showing an inner product space is complete

I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2: Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner ...
1
vote
1answer
34 views

Is projection on a convex closed weakly-sequentially continuos?

I think to have proved the following: Given K a convex closed(maybe also limited is needed)subset(also curve not just subspaces) of an Hilbert space H, is well defined the projection operator $p_K:H ...
1
vote
0answers
47 views

Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
1
vote
1answer
58 views

If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
1
vote
1answer
30 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
1
vote
1answer
26 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
1
vote
0answers
46 views

Applications of uniformly convex and uniformly smooth of Banach space

I am studying on geometry of Banach space, I want know applications of uniformly convex and uniformly smooth of Banach space in some branches of mathematics and engineering. Can you help me Thanks ...
1
vote
0answers
58 views

Exercise on additional regularity of reaction-diffusion equation problem

I have to make exercise 6.2.6.1 form the notes of Lunardi. Prove the following additional regularity properties of the solution to (6.12): (i) if $u_0 \in BUC(\mathbb{R}^n,\mathbb{R}^m)$, then $u(t, ...
1
vote
0answers
43 views

existence of eigenfunction for an operator

I'd like to know whether there's a general condition on an operator for it to have an eigenfunction. For example, differential operator has eigenfunction $f_k (x)=e^{kx}$ , and differential operator ...
1
vote
1answer
43 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
1
vote
0answers
39 views

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Suppose $u \in L^2(0,T;L^2)$, $u_t \in L^2(0,T;H^{-1})$ and $f \in L^\infty((0,T)\times\Omega)$. I have the weak form $$\langle u_t, \varphi \rangle_{H^{-1}, H^1} + \int_\Omega\nabla u \nabla \varphi ...
1
vote
0answers
68 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
1
vote
1answer
19 views

convergence in measure implies the composition of the sequence of functions and a continuous function also converges in measure

Let $D$ be a measureble set in $\mathbb{R}^n$. Suppose $\mu(D)<\infty$. Let $\phi: D\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that for almost every $x\in D$, ...
1
vote
1answer
33 views

$\sup_t |T(t)|<+\infty$ implies $\sup_t |T(t)^*|<+\infty$?

Let $X$ be a Banach space. $T(t)$ a family of bounded operators for $t\in\mathbb{R}$. $T(t)^*$ is the adjoint operator of $T(t)$. If $\sup_t |T(t)^*|<+\infty$ , then by Hahn-Banach, there's a ...
1
vote
0answers
26 views

Convergence of cadlag functions to a continuous one

I want to prove a convergence result as simple as possible. Using a straight forward approach I can prove the result, but I'm 100% sure that there must be a much simpler (and shorter) argument using ...
1
vote
0answers
22 views

Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in ...
1
vote
0answers
22 views

Complexity of a Borel linear subspace of a Banach space

This question is inspired by the MO question Image of $L^1$ under the Fourier transform, but I think it might be much easier so I am posting it here instead. Let $(X, \|\cdot\|)$ be a separable ...