Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
3
votes
2 answers

Creating a reflexive Banach space from a weakly compact set.

I have a set $K$ which is weakly compact in a Banach space $E$. Also, its span is norm-dense in $E$. ($E$ is weakly compactly generated. But this doesn't enter the question anywhere.) I need to construct a reflexive Banach space, so I need its…
roo
  • 5,598
3
votes
1 answer

Identity Operator from $( c_{00} , \| \cdot\|_1)$ to $( c_{00} , \| \cdot\|_2)$

Consider the normed vector space $c_{00}(\mathbb N ) = \{ x=(x_n) \in \mathbb{R} ^{\mathbb N} : \{ x_n \neq 0 \} \quad \text {is finite} \}$. Let $Id$ the identity operator. Let $ X =( c_{00} , \| \cdot\|_1)$ and $ Y= (c_{00} , \| \cdot\|_2)$, where…
passenger
  • 3,793
3
votes
0 answers

Dual space of L infty space

The Dual space of L infty space is not L1 ,are there some example to show this? I am going to use rieze representation thm,but it can not be used because p= $\infty$.
mnmn1993
  • 435
3
votes
2 answers

Functional Analysis: Folland Problem

Folland problem 5.36 (b) Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbb{N}$. Suppose that $\{x_{n}\}_{n=1}^{\infty}$ is a countable dense subset of the unit ball of $\mathcal{X}$, and define…
user24367
  • 1,286
3
votes
0 answers

consider the normed linear spaces $(\mathcal C[0,1], ||.|| _i)$.what can you conclude about the correspoding open unit balls?

consider the normed linear spaces $$(\mathcal C[0,1], ||.|| _1), \;(\mathcal C[0,1], ||.|| _2),\;(\mathcal C[0,1], ||.|| _3)\ldots, (\mathcal C[0,1], ||.|| _p)$$ and $(\mathcal C[0,1], ||.|| _\infty)$. Then what can you conclude about the…
David
  • 4,201
3
votes
2 answers

extend a linear function

Let $P$ denote the subspace of $C^0([0,1])$ defined by polynomials restricted to [0,1]. Suppose that $l:P\rightarrow \mathbb{R}$ is a linear function with the property that $p(x)\geq 0$ in $x\in [0,1]$ implies $l(p)\geq 0$. Then how can we show…
violin
  • 173
3
votes
2 answers

Is this some kind of adjoint?

I have a bounded operator $T$ from $L^p$ to itself for $1 \leqslant p \leqslant \infty$. Furthermore, on $L^2$ we have that $T$ is self-adjoint. Now I wish to relate $\|(Tf)g\|_{L^1}$ to $\|f(Tg)\|_{L^1}$ (equal up to a constant perhaps). What…
JT_NL
  • 14,514
3
votes
1 answer

Definitions for L2 and Lp Spaces?

I am taking a course in Functional Analysis online, and unfortunately some important terms have not been well defined. In particular, isn't L2 space just Lp space with p=2 ? If so, why aren't continuous functions on closed intervals with the L2…
PossumP
  • 1,734
  • 13
  • 24
3
votes
2 answers

A reflexive Banach space is separable iff its dual is separable

Let $(X,||\cdot||)$ be a reflexive Banach space. Prove that $X$ is separable if and only if $X'$ (the dual space of $X$) is separable. Does anyone have a hint for me? I have no idea where to begin
PublicRecord
  • 45
3
votes
0 answers

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. $\int_{|f| \geq M} |f|\,dm \leq \epsilon\ \forall f \in \mathcal{F}$. 2) A…
Singhal
  • 940
3
votes
1 answer

Closed map on Banach Space

Let $X,Y$ be Banach spaces and $T \in B(X,Y)$. Show that if $T$ sends every bounded closed subsets of $X$ onto closed sets of $Y$ then $T(X)$ is closed. It's true when the map is injective, but is it true in general?
Arindam
  • 835
  • 7
  • 18
3
votes
1 answer

norm of bounded linear operator restricted to dense subspace

I have no idea how to do this question I was given in class. Let $E$ and $F$ be normed spaces and let $T \in \mathcal{L}(E,F)$. Suppose that $E_0 \subseteq E$ is a dense subspace. Show that $\parallel T_{E_0} \parallel = \parallel T \parallel$. I…
Victoria
  • 307
3
votes
0 answers

Question about proof in functional analysis book

I'm currently working through introductory functional analysis from kreyszig, and I don't quite understand one of the proofs. this is it: My question is why it is necessary to go from $k = 1, 2, ...$ to the infinite sum. Why can't we just let…
user2520938
  • 7,235
3
votes
2 answers

SOT stronger than WOT - looking for a proof

I am working on a homework problem to which we were given a hint: "use the fact that the Strong operator topology is stronger than the Weak operator topology". The setting is this: Let $E,F$ be normed spaces, $\phi:B(E,F)\to \mathbb{F}$. Define…
roo
  • 5,598
3
votes
1 answer

Mean Value Theorem for Functionals

The following is paraphrased from Introduction to integral equations with applications by Abdul Jerri: Let $F$ be a functional with domain $D$. Start with $F(x,t,u(t))$. If $F$ has a continuous partial derivative $\partial F / \partial u$ in $D$,…
Matt
  • 2,343
1 2 3
…
99 100