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
4
votes
1 answer

Why the Jauge of $C$ is $\inf\{r>0\mid v/r\in C\}$ and not $\sup\{r>0\mid rv\in K\}$?

Why the Jauge $p:\mathbb R^n\to \mathbb R$ of a set $C\subset \mathbb R^n$ is defined as $$p(v)=\inf\{r>0\mid v/r\in C\}$$ and not as $$p(v)=\inf\{r>0\mid rv\notin C\} \quad \text{or}\quad p(v)=\sup\{r>0\mid rv\in C\}\ \ ?$$ Because both explain the…
user330587
  • 1,624
4
votes
3 answers

Why is $d(x,0)$ not a norm?

If $\|x\|$ is a norm, then we can define $d(x,y):=\|x-y\|$ and it will be a metric. Now, if $d$ is a metric, why is $\|x\|:= d(x,0)$ not a norm? I think it fail for the sub-linearity, but I don't see how.
Peter
  • 1,005
4
votes
1 answer

Closed set in metric space

In $C\left( {\left[ {0,1} \right]} \right)$, we define metric $$d\left( {f,g} \right) = \mathop {\max }\limits_{t \in \left[ {0,1} \right]} t\left| {f\left( t \right) - g\left( t \right)} \right|.$$ We define $A = \left\{ {f \in C\left( {\left[…
Trần Quang Minh
  • 431
4
votes
1 answer

Finding a certain sequence for a discontinuous linear map

Let $X,Y$ be normed spaces and $T:X\to Y$ linear and discontinuous. Hence $T$ is discontinuous at every point. Then for every $x\in X$ there exists a sequence $(x_n)\subseteq X$ such that $(x_n)$ converges to $x$ and $(T(x_n))$ doesn't converge to…
Tanius
  • 1,195
4
votes
1 answer

Is weakly compact subset weakly separable?

Let $X$ be a Banach space and $K$ be a weakly compact subset of $X$, i.e. compact set with respect to the $\sigma(X, X^{*})$-topology. Is it true that $K$ is weakly separable? Since weak topology is never metrizable for infinite dimensional space,…
Seewoo Lee
  • 15,137
  • 2
  • 19
  • 49
4
votes
1 answer

self-adjoint operator with positive spectrum is positive

I have found many threads about the inverse statement but none for this: (Part of exercise 6.24 in Brezis) Let $T$ be linear, self-adjoint operator on a Hilbert space and assume that the spectrum of $T$ fulfills $\sigma(T) \subset [0,\infty)$. Prove…
Philipp Wacker
  • 431
  • 2
  • 13
4
votes
3 answers

consequence of Hahn-Banach theorem

In Wikipedia, it says that Hahn-Banach Theorem shows there are "enough" continuous linear functionals. But, why is that so in a space that is not necessarily normed? How does the statement of Hahn-Banach show this?
nan
  • 719
  • 4
  • 15
4
votes
2 answers

A corollary of Hahn–Banach theorem and a generalized limit function of $\ell_\infty$

A corollary of Hahn–Banach theorem states that Let $E$ be a normed vector space, $M$ a proper closed subspace and $x \in E$. If $d(x,M) = \delta > 0$, so exists $f \in E'$ such that $\|f\|=1$, $f(x)=\delta$ and $f(m)=0$ $\forall m \in M…
H R
  • 461
4
votes
2 answers

Find continuos function $x$ with $x(0)=0$ such that $\|x-y\| \geq 1$ where $y(0)=0$ and $\int_0^1 y(t) dt = 0$

Consider $C[0,1]$ space of continuos functions with the uniform norm. Let $X = \{ x \in C[0,1]: x(0)=0\}$ and $Y = \{ y \in X: \int_0^1 y(t) dt = 0 \}$ subspaces of $C[0,1]$. How can I show that $\exists x \in X$, $\|x\|=1$, such that $$\|x-y\|…
H R
  • 461
4
votes
0 answers

Invariant spaces for a shift semigroup

Consider the space $C_0 := \{f \in C[0,1], f(0)=0\}$ and the shift semigroup $(T^t, t \in [0,1])$, defined by $T^t f(x) := \begin{cases} f(x-t), & t\le x\\ 0, & \text{otherwise} \end{cases}$ Clearly, for every $t \in [0,1]$ there is a $T$-invariant…
Alexander Shamov
  • 4,457
  • 17
  • 28
4
votes
2 answers

If $f\in L^1[0,1]\cap L^2[0,1]$, then $\|f\|_1 \le \|f\|_2$.

This problem has two parts: (a) If $f\in L^1[0,1]\cap L^2[0,1]$, then $\|f\|_1 \le \|f\|_2$. (b) Use (a) to deduce that $L^2[0,1]$ is a subset of $L^1[0,1]$. Without using part (a), let $f$$\in$$L^2[0,1]$. Since the constant function $1$ $\in$…
user45955
  • 1,055
  • 9
  • 16
4
votes
2 answers

Bounded linear operator (between Banach spaces) with second category range has closed range

I'm attempting a problem about closed range of a bounded linear operator. Assume $X, Y$ are Banach spaces and $A$ is a bounded linear operator. If $\operatorname{Ran}(A)$ is of the second category, then show that $\operatorname{Ran}(A)$ is…
nekodesu
  • 2,724
4
votes
1 answer

Spectral theorem for Fourier transform

The Fourier transform restricts to an isometry $F$ of the subspace of real even functions in $L^2(\mathbb{R}^n)$. Also $F$ is self adjoint since $$\int F(u)v = \int u F(v)$$ for real even functions. If this is correct, how does the spectral theorem…
alesia
  • 173
4
votes
1 answer

Questions about operator norm

I'm reading about functional analysis and I found the definition of the operator norm, if you have $(X,\|\|_1)$ and $(Y,\|\|_2)$ normed spaces then the set $\mathcal{L}_{\|\|_1,\|\|_2}(X,Y) := \{T:X \to Y \text{ linear }: \sup\{ \|T(v)\|_2: \|v\|_1…
Diego Silvera
  • 1,717
4
votes
1 answer

Give an unbounded linear functional on Hilbert space

We want to use that every Hilbert space has a Hamel basis (where a Hamel basis of $H$ is a set $V \subseteq H$ such that $V$ is linearly independent and such that every element of $H$ can be written uniquely as a finite linear combination of V's…
PBJ
  • 961
1 2 3
…
99 100