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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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
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0answers
30 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
57 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
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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
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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
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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
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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?
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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 ...
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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} = ...
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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 ...
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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
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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 ...
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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
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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 ...
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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
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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
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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, ...
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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
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1answer
24 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
30 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
22 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
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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 ...
1
vote
0answers
32 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
4
votes
1answer
40 views

Sturm-Liouville problem and periodic boundary conditions

I was wondering about this: I know that if a 1-d Sturm-Liouville operator is limit circle or limit point then the eigenvalues are simple ( so no degenerated spectrum). But in the case of periodic ...
3
votes
0answers
32 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
2
votes
0answers
44 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
0
votes
1answer
12 views

Existence of adjoint operator in Euclidean space

If we define the adjoint operator of linear operator $A:E\to E$, where $E$ is a complex or real Euclidean, $n$- or $\infty$-dimensional, space, as operator $A^\ast:E\to E$ such that $\forall x,y\in ...
4
votes
1answer
91 views

Condition on vector-valued function

Does anyone have any ideas on how to show that the following is true: Let $\Omega \subset \mathbb{R}^{n}$ be open and bounded. Consider vector-valued function $$f: \Omega \times \mathbb{R} \times ...
0
votes
1answer
39 views

The cone over separable C*-algebra is also separable?

For a C*-algebra $A$, the cone over $A$ is $CA=C_{0}(0,1]\otimes A$ , My question: If $A$ is separable, $CA$ is also separable?
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1answer
40 views

Two questions about orthogonal projections on Hilbert space

Let $l_{k}^{2}$ denote the k-dimensional Hilbert space and $\oplus_{1}^{\infty} l_{k}^{2}$ be the infinite direct sum of $l_{k}^{2}$. Let $P_{M}\in ...
2
votes
0answers
34 views

Generalization of the Riesz-Markov theorem

So, my professor mentioned a version of the Riesz-Markov theorem for some kind of general spaces, that yields a maximum and a minimum measure rather than a unique measure (or something along those ...
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vote
1answer
15 views

Is it sufficient to check weak convergence on a (weak* or strongly) dense subset of the dual?

Let X be a Banach space. If $D \subset X^*$ is (weak*ly or strongly?) dense, then does $f(x_n) \to f(x)$ $\forall f \in D$ imply that $x_n \to x$ weakly? My thoughts: If $g_m \to g$ in the dual, then ...
1
vote
1answer
20 views

Joint spectrum of $\{a_1,…,a_n\}$

Let $\{a_1,...,a_n\}$ be commuting normal operators on a Hilbert space. Put $A:= C^*(1,a_1,...,a_n)$. By Gelfand theorem ,abelian C*-algebra $A$ is identified with the algebra $C(\Omega)$ of all ...
1
vote
1answer
26 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
0
votes
1answer
21 views

adjoint of composition of bounded linear operators

Our lecturer said it shouldn't be any problem to prove this on our own, but I must be missing something obvious! Let $E$, $F$ and $G$ be normed spaces. Let $T \in \mathcal{L}(E,F)$, $S \in ...
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0answers
14 views

deduce open mapping theorem from Banach's homomorphism theorem

We have been given this question in our homework, and I really just don't have an idea where to start Use the canonical factorisation to deduce the Open Mapping Theorem from Banach's Homomorphism ...
4
votes
1answer
28 views

The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
2
votes
1answer
29 views

Coercivity definitions

Hi I was given the following definition of coercivity: Let $V$ be a Banach space. The first definition: $A:V \rightarrow V^{*}$ is coercive iff $\exists \zeta: \mathbb{R}^{+} \rightarrow ...
0
votes
1answer
43 views

Definition of $C^1, $the vector space of continuously differentiable functions

I asked a question on clarification of the symbol $C^k$. It was confirmed to me that $C^k$ is actually a space of functions. Now my next question in the definition is on $C^0$ and $C^1$. $C^0$ is ...
0
votes
2answers
34 views

Notation for the vector space of functions with $k$ continuous derivatives

I saw the following definition given at the mathworld web site: A function with $k$ continuous derivatives is called a $C^k$ function. In order to specify a $C^k$ function on a domain $X$, the ...