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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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Open subsets of the space of linear operators

If we have a Banach space $X$ and we consider the space $L(X,X)$ of linear operators. Now we have the operator norm here and this induces a metric, which in turn induces the topology. Since this is a metric space, we have the notion of open sets. So…
Atherton
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Compact subsets of Banach spaces

Is a theorem of Mazur that a closed subset $K$ of a Banach space $X$ is compact if, and only if, there is a sequence $(x_n)$ of $X$ such that $|x_n|\to 0$ and $K\subseteq \overline{\operatorname{conv}}\{x_n:n\geq 0\}$, where…
user34870
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Proving that a linear operator $T$ is continuous.

If the space $X$ is banach , then I want to show that any linear map $T:X \to X$ is continuous iff the null space is closed. I could show that if $T$ is continuous then the null space is closed. But I am unable to prove the converse. Any hints are…
happymath
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show that closure of $c_{00}$ is $c_{0}$ in $\ell$-infinity

Let $x=(x(1),x(2),…,x(n),…) \in c_0$. So for any $\varepsilon>0$ there exists a $n_0 \in \mathbb{N}$ such that $|x(n)| \to 0$ as $n \to \infty$ for all $n \geq n_0$. Now for all $n \geq n_0$ , $||x_n − x||_\infty = \sup \{|x(m)|: m \geq n_0 \} \to…
sarani
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Absolute Convergence of a Series defined as a Cauchy Sequence

So the question I'm answering is "Suppose (X, || ||) is a normed space. Show that X is complete iff every absolutely convergent series in X converges on an element of X." The first half was simple (show that every absolutely convergent series can be…
Iceman
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To show a dense subspace of lp(Z)?

Exercise: My proposal Recall that \begin{equation} \|u\|_p := \left( \sum\limits_{j \in \mathbb{Z}} | u_{j} |^{p} \right)^{1/p}, \,\,\, j \in \mathbb{Z}, \forall u = \left( u_{j} \right) \in l_{p}, \, u_{j} \neq 0 \, \text{ if } 1…
Léo Léopold Hertz 준영
  • 439
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$L^2([0,1])$ is a set of first category in $L^1([0,1])$?

How to show that $L^2([0,1])$ is a set of first category in $L^1([0,1])$? Thank you.
Tom
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$\ell^p$ as a direct summand of $L^p$

I've been struggling with the following problem from a previous year's quals, and I don't know where to look it up (or even if it's supposed to be too obvious to write down). How do we embed $\ell^p$ as a direct summand of $L^p(0,1)$? In other…
DCT
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Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

Problem Let $f_1,f_2,\ldots, f_n$ be linear functionals on a vector space $X$. Show that there exist constants $\lambda_1,\ldots,\lambda_n$ satisfying $$f=\sum_{i=1}^n\lambda_i f_i$$ if and only if $\bigcap_{i=1}^n \ker f_i \subset \ker…
recmath
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A continuous mapping with the unbounded image of the unit ball in an infinite-dimensional Banach space

Let $X$ be an infinite-dimensional Banach space, and let $B=\{x\in X: \|x\|\leq 1\}$ be the closed unit ball of $X$. Please give an example of a continuous mapping $F: X\to X$ such that $F(B)$ is unbounded.
Anastasia V
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Bounded inverse operator

Let $X$ and $Y$ be Banach spaces. Suppose $T$ is a linear operator from $X$ onto $Y$ with $\operatorname{Dom}(T)\subset X$. Show that $\exists T^{-1}\in L(Y,X)\Leftrightarrow\exists M>0:\ \left\Vert x\right\Vert \leq M\left\Vert Tx\right\Vert…
user16859
  • 705
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For a self-adjoint $T$, if $T^k$ is compact, then so is $T$.

I'm studying functional analysis. Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a bounded self-adjoint linear operator on a Hilbert space $\mathcal{H}$. The problem is showing if $T^k$ is a compact operator for some $n\in\mathbb{N}$, then $T$ is also…
D. Lee
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How do I prove that there doesn't exist a unit norm vector at a unit distance from a closed subspace of an infinite dimensional vector space?

Let $M$ be a proper closed linear sub space of a normed linear space $X$. If $X$ is finite dimensional, it's a well known result by F.Riesz that there exists a unit vector $x$ such that dist($x,M$)=$inf_{m\in M}\|x-m\|=1$. This need not be true if…
pikachuchameleon
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If $T: \ell^2 \to \ell^2 $ is defined as $Tx = (\frac{x_i}{i})$, find $\lVert T \rVert$.

Here $(x_i) \in \ell^2$. All I have on paper right now is $$\rVert Tx\rVert = \sqrt{\sum_{i=1}^{\infty} \frac{x_i ^2}{i^2}}$$ and I'm not sure about what's next. Please help.
Una191
  • 131
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If $M$ is a non-empty subset of a Hilbert space $H$, the span of $M$ is dense in $H$ iff $M^{\perp} = \{0\}$.

I need help only with the converse of the proof. Suppose, $M^{\perp} = \{0\}$, and $V=span(M)$, then if $x \perp V$, this implies $x \perp M$ (why?) so that $x \in M^{\perp}$ and $x=0$. Hence $V^{\perp} = \{0\}$ (again why?). Noting that $V$ is a…
Una191
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