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)

0
votes
2answers
27 views

Any example of non-closed operator?

I cannot think of one. By the way, is there any good exercise book on functional analysis or hilbert space?
0
votes
1answer
18 views

Is Hlawkas Inequality holds for sobolev space

im wondring is that inequality holds for any functionnal space such as sobolev space and if it's true how we can write it in that space /HlawkasInequality any help would be apperciated
1
vote
0answers
27 views

No norm consistent with given topology

Given the (Frechet) topology on the Schwartz class $S(\mathbb{R}^d)$ induced by the seminorms $\rho_{\alpha \beta}f = \operatorname{sup}_{x \in \mathbb{R}^d}|x^{\alpha}\partial^{\beta}f|$, how can I ...
0
votes
1answer
21 views

Smooth and Lipschitz domains

We know that an open ball $B_{r}\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. How can I show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is ...
1
vote
1answer
38 views

Unit sphere weakly dense in unit ball

I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball. I already had to prove that the unit ball ...
2
votes
1answer
32 views

A basic question on $L^p$ norm

Consider a probability space and $f_m$ be sequence of measurable functions a.s. converging to $f$. What can be said about the limit $$ \lim_{m\to \infty} \|f_m\|_m$$ where $\|.\|_p$ stands for the ...
0
votes
1answer
31 views

Position Operator on $l^2(\mathbb{Z})$

I'm very familiar with the position operator $(Q\varphi)(x)=x\varphi(x)$ on $L^2(\mathbb{R^d})$, but I'm trying to figure out how to interpret the same operator on $l^2(\mathbb{Z})$ (the space of ...
0
votes
1answer
18 views

Multilateral shift operators

Apparently, the operators $u(e_n) = e_{n+1}$ and $u^\ast (e_n ) = e_{n-1}$ are called "unilateral shift oeprator". Since they have to be called that (instead of calling them "the shift operator") ...
0
votes
0answers
12 views

Let $B$ be the Banach space of complex measures, with norm $|\mu|(X)$. What is the Banach space interpretation of notions such as mutual singularity?

Let $B$ be the Banach space of complex measures, with norm $|\mu|(X)$. What is the Banach space interpretation of measure theoretic notions such as mutual singularity? Is there a useful dictionary ...
1
vote
1answer
89 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
1
vote
1answer
14 views

Approximation in $V^{**}$

We know that for a vector space $V$ there is a natural map $\iota\colon V\to V^{**}$ sending $v$ to $v^{**}$. When $V$ is a normed space, $\iota$ is an isometry, however it may not be surjective. My ...
1
vote
1answer
32 views

geometric interpretation of analytical hahn-banach theorem

I understand this interpretation. But how can I see this in example?
0
votes
1answer
14 views

Is any bounded sequence weakly precompact in a reflexive banach space?

please help me to proov this property: In a reflexive banach space any bounded sequence is weakly precompact
0
votes
3answers
28 views

Why cannot a densely defined operator be extended to an everywhere defined operator?

I am a physicist learning functional analysis because of its fundamental role in quantum mechanics. There are so many bizarre facts. One is, there are densely defined operators which seem cannot be ...
1
vote
1answer
15 views

Why only densely defined operators can have an adjoint operator?

Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?
4
votes
0answers
208 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
3
votes
2answers
53 views

Examples of orthonormal bases for $L^2[0,1]$ that are not trigonometric?

What are examples of orthonormal bases for $L^2([0,1],dx)$? For instance, the following trigonometric polynomials are orthonormal basis $$\left\{1, \sqrt{2}\sin(2\pi jx),\sqrt{2}\cos(2\pi j ...
1
vote
1answer
30 views

embedding of $\prod_{n\in\mathbb{N}}M_{n}(\mathbb{C})$ in a type $II_{1}$ factor

Suppose $M$ is a type $II_{1}$ factor with trace $\tau$. Let $\lbrace p_{n}\rbrace_{n\in\mathbb{N}}$ be an increasing sequence of projections such that $\tau(p_{n})\rightarrow 1$. Now, let's consider ...
0
votes
2answers
12 views

Study of a parametric function

I would like to study this function for $x\geq 0$, $\forall b,d \in \mathbb{R}$: $$ y=\frac{b+dx}{1-b-dx} $$ Can I say that it is monotone increasing (decreasing) over $x$ in its domain for $d>0$ ...
2
votes
0answers
31 views

Showing inequalities for $l^p$ sequences

If I show that an inequality (e.g. Holder or Minkowski) holds for the $L^p$ space, then can I automatically conclude that the inequality also holds for $\ell^p$ sequences, just by integrating wrt. the ...
3
votes
0answers
59 views

Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
1
vote
3answers
123 views

An example of non-closed subspace of a Hilbert space?

I am reading a book on Hilbert space. It seems that the author assumes that a linear subspace of a Hilbert space can be non-closed. I cannot think of an example. I am still used to the ...
1
vote
0answers
26 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
2
votes
0answers
33 views

Uniform integrability and weak sequential precompactness

Over a probability space $( X, \mathcal{B}, m )$, 1) A collection $\mathcal{F} \subset L^1 (m)$ is called uniformly integrable if for all $\epsilon > 0,\ \exists M > 1$ s. t. $\int_{|f| \geq M} ...
0
votes
0answers
18 views

Let $C:A\to B$ be a closed operator such that $\operatorname{Dom}(C)$ is closed. Prove that $T$ is continuous.

Let $A$ and $B$ be two Banach spaces. Let $C:A\to B$ be a closed operator such that $\operatorname{Dom}(C)$ is closed. Prove that $T$ is continuous. Any ideas?
1
vote
0answers
36 views

The paralelogram law and the polarisation identity

I am struggling to remember the parallelogram law and the polarisation identity. Every time I need one of the two I have to look them up both. Therefore I would like to better understand the two so ...
0
votes
1answer
17 views

Adjoints and $\operatorname{im}{(u^\ast)}$: is the orthogonal complement of a closed subspace closed?

Context (you may skip this part of my question): Let $H,H'$ be Hilbert spaces and $u\in B(H)$ and let $u^\ast$ be the adjoint of $u$. It is clear that $\operatorname{ker}{u^\ast} = ...
0
votes
0answers
8 views

Find the upper and lower densities of the sets a) $\{e^{2\pi ix(3n+1)}\}_{n \in \mathbb{Z}}$,

Find the upper and lower densities of the sets a) $\{e^{2\pi ix(3n+1)}\}_{n \in \mathbb{Z}}$, b) $\{e^{\pi i((3n+1)x+(4m+1)y}\}_{n \in \mathbb{Z}}$, and c) $\{e^{\pi i((3n+m)x+(4m+n)y}\}_{n \in ...
0
votes
0answers
15 views

How to finish this proof that $u^\ast$ is linear

As I asked in this comment here: I have shown that for all $h\in H$ where $H$ is a Hilbert space we have $$ \langle h, u^\ast (\lambda h' + \mu h'' )\rangle = \langle h, \lambda u^\ast (h') + \mu ...
0
votes
0answers
13 views

Prove that a complex-valued homomorphism on a Banach algebra which is not identically 0, is a bounded linear functional of norm $1$

I want to prove that a complex-valued homomorphism $h$ on a Banach algebra $X$ which is not identically 0, is a bounded linear functional of norm $1$. This is a statement in the appendix D of the ...
0
votes
1answer
15 views

How to show that $u^\ast$ is linear

I am trying to prove the existence of adjoints of bounded linear operators on Hilbert spaces: If $H,H'$ are Hilbert spaces and $u \in B(H,H')$ then there exists a unique $u^\ast \in B(H',H)$ such ...
0
votes
0answers
10 views

Find $t>0$ ao that the set $\{e^{2\pi ix(n+t\,cos(n))}\}_{n\in \mathbb{Z}}$ is a Riesz bases of $L^2(0,1)$.

Find $t>0$ ao that the set $\mathscr{B}=\{e^{2\pi ix(n+t\,cos(n))}\}_{n\in \mathbb{Z}}$ is a Riesz bases of $L^2(0,1)$. The set $\{e^{2\pi ixn}\}_{n\in \mathbb{Z}}$ is a Riesz basis of $L^2(0,1)$ ...
2
votes
0answers
39 views

Particular $L^p$ space

I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and ...
0
votes
0answers
16 views

Is it true that for every signed probability distribution `f`, there are positive distributions `g` and `h` st. `fg=h`?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem: For every generalized g.f. f (of a signed probability distribution) there exist two ...
1
vote
0answers
24 views

tensor products of Banach space

Let $E_{1},\cdots, E_{n}$ be Banach spaces; $n\in\mathbb{N}$ and $\mathbb{R}$ be a real numbers and $E\widehat{\otimes}\mathbb{R}$ be a completion tensor product. We have the fact that ...
1
vote
0answers
29 views

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
-1
votes
0answers
13 views

Find tight frames of density a) $d=4$, b) $d=9$, c) $d=16$ of $L^2(B)$, where $B$ is the unit circle of $\mathbb{R}^2$.

Find tight frames of density a) $d=4$, b) $d=9$, c) $d=16$ of $L^2(B)$, where $B$ is the unit circle of $\mathbb{R}^2$. Can you find a tight frame of $L^2(B)$ with density $d<4$? How would I find ...
0
votes
0answers
15 views

Gateaux derivative of $f:X \to Y^*$

I have a map $f:X \to Y^*$ between two Banach spaces where $Y^*$ is the dual space of $Y$. How do I calculate $$\lim_{t \to 0}\frac{f(x+th)-f(x)}{t}$$ when $f(x+th) \in Y^*$ and I all I know is how ...
1
vote
1answer
25 views

Analytic vectors of self-adjoint unbounded operators

I am working with self-adjoint unbounded linear operators on infinite-dimensional separable Hilbert spaces, and I would like to know some references regarding analytic vectors, especially with respect ...
-2
votes
1answer
27 views

Justify with reason abot correct option [closed]

Let $f:\mathbb R^2 \to \mathbb R$ be a continuous map such that $f(x)=0$ for only finitely many values of $x$.Which of the following is true? 1.either $f(x)\le 0$ for all x or $f(x)\ge 0$ for ...
0
votes
1answer
32 views

First variation of $\int_\Omega |\nabla u |^p$

How can I calculate the first variation of $F(u) = \int_\Omega |\nabla u |^p$? I cannot expand $|\nabla (u+th)|^p = (|\nabla u |^2 + t^2|\nabla h|^2 + t\nabla u \nabla h)^{\frac p2}$ at all so I ...
0
votes
1answer
31 views

Spectral theorem question

I am trying to understand how to develop the spectral measure of a bounded self-adjoint operator on a Hilbert space. For every continuous function on its spectrum, $f: C(\sigma(A)) \to \mathbb{C}$, ...
1
vote
0answers
24 views

Generator of a Feller semigroup on a coutable space

Let $E$ be a countable set in the discrete topology. Let $(T_t)_{t \geq 0}$ be a Feller semigroup on $E$, i.e. a strongly continuous semigroup of operators on $\mathcal{C}_0(E)$ (in the topology of ...
2
votes
1answer
36 views

Convergence of truncation in $L^{p}$

If you have a truncation $T_{k}u$ defined as: $$ T_{k}u := \begin{cases} u,& \text{ if }~ |u(x)| \leq 1\\ k\frac{u}{|u(x)|}, & \text{ if }~|u(x)| > k \end{cases} $$ If you consider ...
1
vote
1answer
32 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
0
votes
1answer
29 views

Does every continuous operator between normed spaces map bounded sets to bounded sets?

Suppose you have a continuous operator $A$ between two normed spaces $X$ and $Y$. Does it follow that this operator is bounded in the sense that it takes bounded sets to bounded sets, given that $A$ ...
1
vote
2answers
23 views

Summable family in a normed linear space

I learnt a definition: Let $X$ be a normed linear space and $J$ be a non-empty set. A family $x:J\rightarrow X$ is summable with sum $\overline{x}$ if for all $\epsilon>0$, there exists a finite ...
2
votes
1answer
28 views

Operator topologies and examples

In class we covered several operator topologies: the weak topology, the weak* topology, the weak operator topology, and the strong operator topology. The first two are defined on a normed vector ...
0
votes
1answer
17 views

Orthonormal Hamel Basis is equivalent to finite dimension

Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a ...
2
votes
0answers
15 views

Double adjoints and reflexivity

Let $X$ and $Y$ be normed (or Banach) spaces. Does anyone know a nice proof that every bounded operator $T:X \rightarrow Y$ is its own double adjoint (that is $T^{\ast\ast}=T$) if and only if $X$ and ...