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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77 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
3
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1answer
318 views

Compact subset of a Banach space of infinite dimension

Let $X$ be a Banach space of infinite dimension. And let $K\subset M$ be a compact subset of $M$. Can we conclude something about the interior of $M$? It's true that it's empty? I don't know how to ...
3
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1answer
163 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...
1
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2answers
150 views

Sequences and Contraction of a fixed point

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
0
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1answer
248 views

Question about convergence in weak operator topology (from Reed and Simon)

I am reading over Chapter VI in Simon and Reed's Functional Analysis. In the first section, the discussion covers various topologies defined on $\mathcal{L}(X,Y)$, the space of bounded linear ...
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1answer
30 views

How to show equivalence of two definitions of a limit point

In an inner product space $R$, one definition of a limit point of a set $N \Subset R$ is that it is the point $f$ in $R$ which is a limit of a sequence from $N$ (in other words , there is a sequence ...
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1answer
26 views

A Sobolev embedding type question

My question is, if a function is in $H^1(\mathbb{R})$, must it necessarily vanish at infinity? If not, what are some additional criteria required? Thanks.
2
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1answer
738 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
0
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1answer
27 views

Does for $u\in L^1(\Omega)$ and every $t$ also hold $\nabla u \cdot 1_{\{u=t\}}=0$ a.e.?

The result is known if $u$ is more regular e.g. $u \in W^{1,1}(\Omega)$. Is it also possible to extend such an result to mere integrable or even just measurable functions? Unfortunately the result ...
0
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1answer
63 views

Proving that $T(B(x,2\epsilon))\cap B(y,2\epsilon) \neq \emptyset $

$H$ Hilbert space. $x,y \in H$ and $T\in L(H)$ 1) $T(B(x,\epsilon))\cap B(0,\epsilon) \neq \emptyset $ 2) $T(B(0,\epsilon))\cap B(y,\epsilon) \neq \emptyset $ 3) $T(B(x,2\epsilon))\cap ...
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1answer
37 views

Bound on surface gradient in terms of gradient

Let $S \subset \mathbb{R}^n$ be a hypersurface and define the surface gradient of a function $u:S \to \mathbb{R}$ by $$\nabla_S u = \nabla u - (\nabla u \cdot N)N$$ where $N$ is the normal vector. Is ...
2
votes
2answers
86 views

Prove $\int_0^1|f(t)-g(t)|dt \le (\int_0^1|f(t)-g(t)|^2dt)^{1/2} \le \sup_{t\in[0,1]}|f(t)-g(t)|$

Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$. Let these be 3 metrics on $C$. $p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$ $d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$ ...
4
votes
2answers
107 views

non separable implies an uncountable set with lower bounded distances?

Given a Banach space the only way i've seen to show that it is not separable is to show that there is a more than countable set $A$ and a costant $c>0$ such that $|a_1-a_2|>c, \forall a_1 \neq ...
1
vote
1answer
178 views

A question about Riesz - Fischer theorem's proof

In Riesz-Fischer theorem's proof, when we put $$ g_k =|f_{n_1}|+|f_{n_2}-f_{n_1}|+ \cdots + |f_{n_k}-f_{n_{k-1}}| $$ it is easy to get (by Minkowski's inequality) $$ \left \| g_k \right \|_p \leq ...
1
vote
1answer
323 views

Diagonal sequence trick: Uniform bound necessary?

There is a treatment of the "diagonal sequence trick" in Reed and Simon (Functional Analysis Vol.1) stated there as follows: Let $f_n(m)$ be a sequence of functions on the positive integers which ...
2
votes
1answer
68 views

Convergence and Constant sequence?

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0 x_1 x_2...$ given by $x_n = g(x_{n-1})$. converges ...
2
votes
0answers
32 views

$\langle f, u \rangle = 0$ for all $u \in C_c^\infty(0,T;H)$ implies $f=0$? Please check proof

Let $f \in L^2(0,T;H)$ where $H$ is a Hilbert space. Suppose that $$\langle f, u \rangle = 0\quad\text{for all $u \in C_c^\infty(0,T;H)$}$$ where the dual pairing is the one between $L^2(0,T;H)$ and ...
2
votes
2answers
136 views

Isometric identification of $c_0^*$ and $ \ell^1$

Let $\{x_n\}_{n=1}^{\infty}\subset \ell_1$ be a sequence in $\ell_1$ with $x_n = (x_n(1),x_n(2), x_n(3),\ldots )$ I want to show that $$\lim_{n\to\infty}\sum_{j=1}^{\infty} x_n(j)y(j) = 0 $$ for all ...
3
votes
1answer
85 views

Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed

I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's ...
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1answer
79 views

Weak* sequentiality

Suppose we are given a Banach space $E$ such that weak* compact subsets of $E^*$ are weak* sequentially compact (for example this happens when $E$ is separable). Does it follow that if $A$ is a subset ...
2
votes
1answer
105 views

$C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
2
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0answers
70 views

Tensor products of weakly compact sets

Let $X$ and $Y$ be Banach spaces. Denote by $X\otimes_\varepsilon Y$ the injective tensor product of $X$ and $Y$. Also, let $A\subset X, B\subset Y$ be any sets. Set $A\otimes B = \{a\otimes b\colon ...
3
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0answers
99 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
0
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1answer
62 views

Completions of a vector space with inner product

Assume $(H,(\cdot,\cdot)_H)$, and $(G,(\cdot,\cdot)_G)$ are two vector spaces with inner product. Suppose $A:H\rightarrow G$ is a linear isometry. Let $T(H)_{*}$ be the completion of $T(H)$ with ...
4
votes
1answer
143 views

g continuous on [a,b] using intermediate value theorem

Suppose that g is continuous on an interval [a,b] and that $g(x) ∈ [a,b]$ for all $x ∈ [a,b]$. (a) Use the intermediate value theorem to prove that is at least one number $c ∈ [a,b]$ with $g(c) = ...
1
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1answer
312 views

Minimal distance between quadratic function and point

I have a function or line $R\rightarrow R^n$ $$ y_i = f(x) = {-b_i \pm \sqrt{-4 \cdot a_i\cdot c_i + b_i^2+4\cdot c_i \cdot x}\over 2 \cdot c_i}$$ where the parameter vectors $\mathbf{a}, ...
1
vote
1answer
148 views

How to show that the dual of $(\mathbb{R}^n,\|{\cdot}\|_p)$ is $(\mathbb{R}^n,\|{\cdot}\|_q)$?

I am trying to brush up on my functional analysis and I learn some $L_p$ spaces since I was never formally intrduced to them through courses. I wanted to know if anyone could offer me a proof or give ...
0
votes
1answer
58 views

A question about download the recent paper. [closed]

I am interested in the journal about operator theory, such as Studia Math and Operators and Matrices. However, my college do not buy some journals. How can I get the paper from these journal?
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1answer
65 views

differentiability/holomorphicity of family of bounded operators

Edit: It seems I made a mistake in the statements on differentiability. I will replace weak differentiable implies strong differentiable with weak continuously differentiable implies strongly ...
4
votes
1answer
1k views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
3
votes
1answer
65 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...
4
votes
1answer
134 views

Dual space of $C_b(X)$

Due to a previous question I wonder if one knows the dual space of $C_b(X)$. Here $C_b(X)$ is the space of all continuous bouded functions with values in $\mathbb{R}$. Of course this depends on the ...
2
votes
1answer
95 views

$\sum 2^{-n} P_n$ of bounded linear operators always bounded?

I have a Banach space X and infinite sequence of bounded linear operators $P_n$ such that for every of them there exists $x_n$ such that $$ ||P_n(x_n)||=2^{2n}||x_n|| $$ These elements $x_n$ ...
-1
votes
1answer
55 views

Is $f(x,y) = x^2 + y^2 + 2$ coercive?

Is $f(x,y) = x^2 + y^2 + 2$ coercive? Im a little stuck on the idea of $x^2 + y^2 = 0$? Can I use this, or not?
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1answer
56 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
2
votes
3answers
73 views

Operator attains norm

We have the following linear and well-defined mapping $T_{a}(b):=\sum_{j=1}^{\infty}a_{j}b_{j}$, with $\{a_{j}\}\in \ell^{1}$ and $\{b_{j}\} \in c_{0}$. Show that $\|T_{a}\|=\|a\|_{1}$. My work: ...
4
votes
1answer
47 views

continuity of a map on $M(\mathbb{R}^n)$

Let $M:=M(\mathbb{R}^n)$ be the space of probability measures on $\mathbb{R}^n$ with respect to the Borel $\sigma$-algebra. Let $K\subset M$ be a compact convex subset. $K$ carries a natural ...
3
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1answer
88 views

missing a condition for convergence or not

In preparation of a test I have some problems solving the following problem: Let $A:=\{a\in \ell^{2}: \phantom{x} \|a\|_{2}\leq 1\}$, $(a_{n})_{n} \in A, a \in A$. Prove that for all $b \in ...
2
votes
1answer
42 views

Proving that the Fourier coefficients of a functional determine it

Proving that the Fourier coefficients of a functional determine it I have the following exercise, taken from old homework of a functional analysis course: Let $\mu\in C(\mathbb{T})^{*}$. Define ...
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1answer
57 views

A problem of maximization in Banach spaces

Let $X$ be a Banach space and $K\subset X$ compact. Let $C(K)$ be the set of continuous function in $K$ and $\mu\in (C(K))^\star$ a non-negative measure. Assume that $f:K\to X^\star$ is a continuous ...
7
votes
3answers
652 views

What is the relation between weak convergence of measures and weak convergence from functional analysis

To keep things simple, we assume $X$ to be a polish space (think of $X$ as $\mathbb{R}^n$ for example). Let's denote with $P(X)$ the space of all Borel probability measure on $X$. We say ...
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0answers
68 views

A problem with a proof that $L^{p_{2}}\subseteq L^{p_{1}}$ for $1\leq p_{1}\leq p_{2}\leq\infty$

In a functional analysis course I saw a claim that for $1\leq p_{1}\leq p_{2}\leq\infty$we have it that $L^{p_{2}}\subseteq L^{p_{1}}$ I have a few problems with the proof given, and I would ...
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vote
1answer
57 views

Is the space $s$ separable?

There is a question "Prove the space $s$ is not separable".And let $x=(x_{i})_{i=1}^{\infty}$, the space $s$ is a sequence space s.t. $s=\{x\}$ with ...
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0answers
30 views

Finding Borel measures on a closed convex hull

Let $M=C(I)^{\ast}$, the space of complex Borel measures on the unit interval $I$. Suppose we give $M$ the weak*-topology induced by the Banach space $C(I)$. Now $\forall$ $t \in I$, let $e_t \in M$ ...
2
votes
1answer
76 views

Banach-Mazur distance

I am stuck upon the following problem. Consider the Banach-Mazur distance for $X$ ,$Y$ normed isomorphic vector spaces $$d(X,Y) = \inf \{ \| T \| \| T^{-1} \| : T \in GL(X,Y) \}$$ I would like to ...
4
votes
1answer
270 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
0
votes
2answers
191 views

The Fourier series converges absolutely $\implies$ it converges uniformly.

Let $S_N(f)$ be the $N$th partial sum of the Fourier series for $f$. I.e. $$S_N(f) = \sum_{n = -N}^{N} \hat{f}(n) e^{2\pi i n x / L}$$ Suppose that the Fourier series converges absolutely, i.e. ...
2
votes
1answer
59 views

Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
0
votes
2answers
57 views

How do they do this w.l.o.g. so freely (Fourier series).

Theorem 2.1. Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Then $f(\theta_0) = 0$ whenever $f$ is continuous at the point ...
0
votes
1answer
84 views

Discontinuous linear operator e and its core

Let $T:E\to \mathbb{R}$ nonzero linear operator with $E$ vector space. So, show the following equivalence: $T$ is discontinuous if and only if $Ker(T)$ is dense in $E$.