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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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Sequence of linear functionals on $\ell^{\infty}$

I am having trouble with this question. Let $X$ be a normed space and let $T : X\to \ell^{\infty}$ be a bounded linear operator. $a)$ Show that there is a sequence $(f_n)_n$ in $X^*$ so that $Tx=(f_n(x))$ for all $x \in X.$ $b)$ Suppose that $X$ is…
taupi
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Functional analysis and linear homeomorphism

Here's a question in one of our exercises list : Let $E$ be a normed vector space, $F$ be a Banach space and $T : E \to F$ a continuous linear application. Define $$ E/\mathrm{Ker} (T) = \{ [x] \} $$ where $[x]$ is the equivalence class of $x \in…
Patrick Da Silva
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Dense subspace of $\ell_2$

How can I show that the subspace $\{x\in \ell_2 \mid \sum_{n=1} ^\infty x_n \frac 1{\sqrt n}=0\}$ is dense in $\ell_2$? Can it be done with "simple" tools just by using the definition of a dense subspace? or can it be done by using the fact that…
MasterJ
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Does ($C([0,1]), \left\|\cdot\right\|_2)$) has a subspace that is isomorphic to ($L^2([0,1]), \left\|\cdot\right\|_2)$)?

Does ($C([0,1]), \left\|\cdot\right\|_2)$) has a subspace that is isomorphic to ($L^2([0,1]), \left\|\cdot\right\|_2)$)? I think the answer is no but I have no idea to prove it.
Pluviophile
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How to prove a space is not separable?

I have a general question on how to prove a space is not separable. I read some posts on this site and it seems like it suffices to find an uncountable family of pairwise disjoint open sets to prove a space is not separable. (here: The space of…
Kenneth.K
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About linearity in Hahn-Banach extension under strict-convex hypothesis

Let $X$ be a Banach space such that $X'$ is strictly convex. Let $Y$ be a closed proper subspace of $X$. Then: $$\forall \varphi \in Y', \exists!\overline\varphi \in X', (\overline\varphi_{|Y}=\varphi)…
Bob
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A problem regarding functional calculus.

Let $A$ be a Banach Space and $T$ be a bounded operator on $A$. Given that, spectrum of $T$, $\sigma[T]$, is $ F_1 \cup F_2;$ where $ F_1, F_2$ are disjoint closed set in complex plane. Show that there exist topologically complemented subspace…
Timon
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Product of $L^2$ functions

If $ \Omega$ is any open subset of $\mathbb{R}^n$, is it true that the product of two $L^2$ functions over $\Omega$ is also $L^2$ ? What about $L^p $ ?
Oussama Sarih
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Rudin's statement of Hahn-Banach's theorem

Rudin's statement of the Hahn-Banach theorem in Functional Analysis (Theorem 3.2) involves a linear form $f$ defined on some subspace of a real vector space, and which is bounded by a sublinear function, as usual. But the specific hypotheses assumed…
Jack M
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Simple proof of Hahn-Banach in finite dimensional space

Let $E$ be a finite dimensional Banach space and $C \subseteq E$ a non empty and convex set such that $0 \notin C$. I have to use the following steps to prove that $\exists T \in E'$ such that $T(x) \geq 0 \, \forall x \in C$: 1) Let…
user90803
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Existence of a continuous mapping that maps the closed unit ball onto its exterior

I saw the post about a continuous mapping $F: X \to X$ where $X$ is an infinite-dimensional Banach space that maps the closed unit ball of $X$ onto the unbounded set. A continuous mapping with the unbounded image of the unit ball in an…
Yber597
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Prove that if f in $C(X \times Y)$ then there exists functions.

Let $X$ and $Y$ be compact metric spaces. I am trying to prove that if $f \in C(X \times Y)$ and $\varepsilon > 0 $, then there exist functions $g_1, g_2,...,g_n \in C(X)$ and $h_1,...h_n \in C(Y)$ so that $|f(x,y) - \sum_{k=1}^n g_k(x)h_k(y) | <…
Helen
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How to prove that $C^k(\Omega)$ is not complete

Let $\Omega \subset\mathbb{R}^n$ be some bounded domain. And Consider the set of all k-times differentiable functions $C^k(\Omega)$. I want to prove that this set is not complete with the inner product $\langle f,g\rangle=\int\limits_{\Omega}f\cdot…
lightningsnail
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Krein Milman Theorem, why the closed convex hull is needed?

I have a question about the Krein Milman that I'm having difficulty answering. Here is the statement of the theorem. If ${K} $is a non-empty compact convex subset of a locally convex space $ {X}$, then ${\text{ext}\;K\neq\emptyset}$ and…
Padraic
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What is an example of a topological vector space which contains a non-absorbent or non-balanced open convex set around the origin?

There are two equivalent definitions of a locally convex topological vector space. Note that since the vector space is topological, addition by a fixed vector x and multiplication by a fixed scalar r are both homeomophisms, which means it is enough…
Vladimir Sotirov
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