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
6
votes
1 answer

Pointwise convergence on a dense subset of the domain

Let's say that $X$ and $Y$ are vector spaces and to be more accurate $Y$ is a dense subspace of $X$. Furthermore we have that $(f_n)_{n\in\mathbb{N}}$ is a sequence in $X^*$, $f\in X^*$ and that $f_n$ converges pointwise to $f$ on $Y$. Can we deduce…
Perpendicular
  • 479
6
votes
1 answer

Are all normed linear spaces Hausdorff? What about a bounded subset of a normed linear space?

The proof that the dual space of a normed linear space is complete in proposition 5.4 of chapter on Banach spaces (John Conway, functional analysis) consists of restricting the functionals in the dual space to a bounded subset. The subset is claimed…
user2808118
  • 678
6
votes
1 answer

structure theorem for Banach spaces

The following is a theorem in the Banach Algebra Techniques in Operator Theory by Douglas: Here are my questions: Could one come up with some reference (or proof) regarding the remark right after the proof: if $\mathscr{X}$ is separable, then…
user9464
6
votes
1 answer

Surjective unbounded linear operator in Banach Spaces

Open mapping theorem says that bounded linear operator $T: X \to Y$ is an open mapping if $X$ and $Y$ are both Banach and $T$ is surjective. I am wondering what about unbounded linear operators? I guess it is not an open mapping but is there any…
TH000
  • 401
6
votes
2 answers

$\ker f$ is either dense or closed when $f$ is a linear functional on a normed linear space

Let $f$ be a linear functional on a normed linear space $X$. Prove that $\ker f$ is either dense or closed in $X$. Two possibilities can occur, i.e either $f$ is bounded or unbounded. If it is bounded then $f$ is continuous and hence $\ker…
Learnmore
  • 31,062
6
votes
1 answer

Understanding why the positive bounded linear functionals on $C_0(X)$ are given by integration against finite Radon measures.

Folland says in his chapter on the dual of $C_0(X)$ (continuous functions on $X$ vanishing at $\infty$): We recall that for any LCH space $X$, $C_0(X)$ is the uniform closure of $C_c(X)$, and hence if $\mu$ is a Radon measure on $X$, the functional…
user21725
6
votes
1 answer

Little $\lambda$-Lipschitz functions

I've read in a few papers that for $\lambda \in (0,1)$ the set of functions given by $$c^{0,\lambda}([0,1]) = \lbrace f:[0,1]\to\mathbb{R}: \lim_{x\to y} \frac{|f(x)-f(y)|}{|x-y|^{\lambda}} = 0 \rbrace $$ is a closed subspace of…
Richard Zhou
  • 61
6
votes
3 answers

Bound of norm of the operator $T(f)=fg$ on $L^p$ space

Prove that: if $g \in L^{\infty}$, the operator $T$ defined by $Tf = fg$ is bounded on $L^{p}$ for $1\leq p\leq \infty$. Its operator norm is at most $||g||_{\infty}$, with equality if $\mu$ is semifinite, where $\mu$ is the measure on…
user24367
  • 1,286
6
votes
1 answer

Is the countable direct sum of reflexive spaces reflexive?

Let $(X_n)$ be a sequence of reflexive spaces. We define the $\ell_2$-direct sum $\bigoplus_n X_n$ as the normed space with elements $(x_n)\in \prod_n X_n$ such that $$ \|(x_n)\|=\left(\sum_{n}\|x_n\|^2\right)^{\frac{1}{2}}<\infty. $$ Is…
Anguepa
  • 3,129
6
votes
1 answer

Why approximative spectrum of operator is closed

Let $A$ be a linear operator on Hilbert space $H$. We say that $\lambda \in \mathbb{C}$ is in approximative spectrum of $A$ iff there exists a sequence $(x_n)$ of vectors such that $\|x_n\|=1$ and $\|Ax_n-\lambda x_n\|\rightarrow 0$. Equivalently,…
Richard
  • 4,432
6
votes
1 answer

Subspace of $C_0(\mathbb R)$ contained in $L^2(\mathbb R)$

Let $W$ be a closed subspace of $C_0(\mathbb R)$ which is continuously contained as also a closed subspace of $L^2(\mathbb R)$. That is, there are constants $c_1$, $c_2$ such that $c_1 \|f\|_\infty \leq \|f\|_2 \leq c_2 \|f\|_\infty$ for all $f\in…
user15464
  • 11,682
6
votes
2 answers

Operator in fractional spaces

I am no mathematician so please excuse me if I don't use the right terminology. Lets say I have a function $y=f(x): x \in \mathbb{R}^3$ and $y \in \mathbb{R}^1$. So this function "projects" points from a higher dimensional space into a lower…
elyase
  • 69
5
votes
1 answer

Collection of all bounded linear operators from $X$ into $Y$ is a normed linear space and is a Banach space if $Y$ is a Banach space

Let $X$ and $Y$ be normed linear spaces, and let $B(X,Y)$ be the collection of all bounded linear operators from $X$ into $Y$ with the operator norm. Show that $B(X,Y)$ is a normed linear space, and $B(X,Y)$ is a Banach space if $Y$ is a Banach…
user3784030
  • 1,013
5
votes
0 answers

Show that $S^{\perp \perp} \equiv (S^\perp)^\perp$ is the closure of $S$.

Suppose $S$ is a (not neccessarily closed) subspace of a Hilbert space $H$. Show that $S^{\perp \perp} \equiv (S^\perp)^\perp$ is the closure of $S$. I know that if $X\in H$, that $X^\perp$ is a closed subspace, but not really sure where to go from…
user3784030
  • 1,013
5
votes
2 answers

In a Hilbert space $H$, if the closed unit ball is compact, then how can it be proved that $H$ is finite-dimensional?

In a Hilbert space $H$, if the closed unit ball $\{x\in H\colon \|x\|\leqslant 1\}$ is compact, then how can it be proved that $H$ is finite-dimensional?
cps
  • 107
  • 1
  • 7
1 2 3
…
99 100