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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
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Who can help me estimate the operator norm of this integral operator?

The operator $T_F(x)$ depending on the function $F\in L^1\cap L^2$ and the real number $x$ is formally defined as follows: $$ T_F(x)\psi(y)=\int_0^{\infty}\psi(t)F(x+t+y)dt $$ Now my question is: Firstly, is this operator well defined as an…
Xuxu
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The Banach-Steinhaus theorem for seminormed spaces

Assume that we have a vector space $X$ over reals with a countable sequence of seminorms $p_n$ on $X$ such that: $$ p_n(x)\leq p_{n+1}(x) \textrm{ for } n\in \mathbb N, x\in X, $$ $$ \textrm{ for } x\in X\setminus \{0\} \textrm{ there is } n\in…
A.B
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What is wrong with the example I gave to contradict a theorem that claim that the closure of $B$ is the unit ball?

I am studying functional analysis, and I saw the following claim Let $X$ be a vector space, $B\subseteq X$ is convex, symmetric around $0$ and s.t $$\bigcup_{n=1}^{\infty}nB=X,\quad\bigcap_{n=1}^\infty \frac{1}{n}B=\{0\}$$ then …
Belgi
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Finding functions $g_{n}$ with norm $1$ s.t $|\phi(g_{n})|\to||\phi||$ where $\phi(g)=\int_{0}^{1}fg$ for some fixed $f\in C([0,1])$

I am trying to do the following exercise: Let $f\in C([0,1])$ and define a functional on $C([0,1])$ by $$ \phi(g)=\int_{0}^{1}fg $$ Prove that $\phi$ is is linear and bounded and find functions $\{g_{n}\}$ s.t $$…
Belgi
  • 23,150
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closed bounded equivalent to compactness?

In preparation for an exam I am struggling with the following problem. We let $A:=\{x=(x_{n})_{n}\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$ and consider $d:A\times A\rightarrow \mathbb{R}_{+}$ defined…
Lech121
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problem metric involving sequences

UPDATED: In preparation for an exam I am struggling with the following problem. We have $A:=\{x=(x_{n})_{n}\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$. Consider the metric $d:A\times A \rightarrow \mathbb{R}_{+}$ defined by …
Lech121
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The Space $\boldsymbol{ba(\mathcal{P}(\mathbb{N}))}$

$\boldsymbol{ba(\mathcal{P}(\mathbb{N}))}$ consisting of all bounded, finitely additive set functions defined on the $\sigma$-algebra $\mathcal{P}(\mathbb{N})$ of all subsets of the natural numbers. Thus, if $\mu: \mathcal{P}(\mathbb{N})\to…
Laplacian
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Why is any norm-bounded family on a reflexive Banach space relatively weakly compact?

Why is any norm-bounded family $T \subseteq L(X)$ on a reflexive Banach space $X$ relatively weakly compact?
Youssef
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Is it legal to chain derivative operators?

I have an infinite summation I'm working with, but there's no 'nice' closed form for the coefficients of this sum without introducing some functionals. But, I'm not sure if the way I've done so is necessarily correct. The coefficients in the sum…
BBadman
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If every subspace of a normed space is closed, then the dimension is finite

Let $(E, \lVert \cdot \rVert )$ a normed vectorial space. If every subspace of $E$ is closed, then $E$ has finite dimension. I have seen this as an excercise of a Functional analysis book and I am wondering how to prove it. My first thought was to…
Superdivinidad
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continuous evaluation of smooth function

Let's consider a smooth function $f:\mathbb{R}\to \mathbb{R}$ and the quantity $$ q(f):= \frac{|f(0)|}{\lVert f\rVert_{L^1(-1,1)}}. $$ If $q(f)$ is rather large it means that $f$ is relatively small at most points in the interval $(-1,1)$ but not at…
mrry0
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Show that $\text{Im}(f \otimes e^*)=\text{span}(f)$

I will start with a definition: Let $E$ and $F$ be Hilbert spaces. For $e \in E$ and $f \in F$ we define the rank-1-operators $(f \otimes e^*)(x):= \langle x,e\rangle f$ for $x \in E$. I want to show the following: $\text{Im}(f \otimes…
Philip
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Question about weak and strong convergence

I have the following exercise that concerns me: Let $X$ be a normed space and $X^*$ its dual space. Suppose $\text{s-lim}\ l_n = l$ and $x_n \to x$. Then $l_n(x_n) \to l(x)$. Does this still hold, if $\text{s-lim}\ l_n = l$ and $x_n \rightharpoonup…
user1265841
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WOT convergence implies SOT convergence of sequence of self adjoint operators. True or False?

I don't believe this to be true but i am having trouble in finding a counterexample of this: let $T_n$, $n \in \mathbb{N}$ be a sequence of self adjoint operators in a Hilbert space $H$ that converges in the WOT topology to $T$. Show that $T_n$…
d.polv
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$X$ is reflexive $\iff$ the transpose map is an isometric isomorphism

I don't understand the following exercise; We are supposed to show that a normed linear space $X$ is reflexive if and only if the transpose map $\;t: B(X,X) \to B(X^*, X^*)$ is an isometric isomorphism. I could already show the first implication,…
Robin Helmig
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