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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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Relation between weak topology and topology of weak convergence?

From Wikipedia, the weak topology on a topological vector space $X$ is the initial topology with respect to its continuous dual $X^*$. In other words, it is the coarsest topology (the topology with the fewest open sets) such that each element of…
Tim
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compact convex problem

Would you please help me in how to show that: If $T : X \to Y$ is a continuous linear map between two locally convex spaces and $K \subset X$ is compact convex, then: The continuous image of compact convex space is a compact convex. If $y$ ia an…
saba
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A Banach space that satisfies parallelogram law is a Hilbert space

A Banach space that satisfies parallelogram law is a Hilbert space. I know the definitions of the space and also the law, but have no idea about how to use it to prove this fact. Thanks in advance for help!
user422112
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Extreme points and closure of a set in $l^2(\mathbb{N})$

Consider the subset C of the banach space $l^2(\mathbb{N})$ defined by \begin{equation} C = \left\{ x \in l^2(\mathbb{N}) \,\Bigg|\, \sum_{n = 0}^{\infty} (n+1) x(n) = 1, \, \, x(n) \geq 0 \, \forall n \in \mathbb{N} \right\} \end{equation} It is…
Tychonoff3000
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find adjoint operator of an operator A

How to find adjoint operator of an operator A $$A \in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$ In answer : for any functional $f_y$ originated by function $y \in BV_0[0,1]:A(f'_y) = g_z$, where functional $g_z$ originated by couple of function…
Gera Slanova
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Functional Analysis: Cannot find a linear combination that involves large scalars but represents a small vector

Theorem: Let $\{x_1,\ldots,x_n\}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c>0$ such that for every choice of scalars $a_1,\ldots,a_n$: $$\left\lVert a_1x_1+a_2x_2+\cdots+a_nx_n…
H_1317
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Cauchy sequences and convergent sequences

I am very confused with a little problem. What is the difference between a Cauchy sequence and a convergent sequence?
Masaud Khan
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kernel and range of a linear operator in a reflexive space

Let $X$ be a reflexive Banach space and $T:X\to X$ is a linear operator. Is it true that $$ X = \mathrm{ker}(I-T) \oplus \overline{\mathrm{range}(I-T^\ast)}, $$ where $\oplus$ is the direct sum? I know this is true when $X$ is a Hilbert space (due…
user58955
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Uniqueness of Hahn-Banach extension

Let $M=\{(x,y,z,0,0,...): x,y,z, \in \mathbb{K}\}$ be a subspace of $(l_p(\mathbb{N}),||\cdot||_p)$ I already proved that $f:M\rightarrow \mathbb{K}$ where $f(x,y,z,0,...)=x-y-z$ is bounded with $||f||=3^{\frac{p-1}{p}}$ for $1 \leq p < \infty$. I…
user289143
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Why should K be closed to ensure X/K is complete?

If $K$ is a closed subspace of Banach space $X$, then $X/K$ is complete. But I think the usual proof of this theorem doesn't make use of the fact that $K$ is closed. Would anyone explain it to me? Thanks a lot.
Zeng
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a ''stronger'' version of Riesz Lemma

Here's a link to a proove of Riesz's Lemma. I think I understand the proof, but can't answere my following question: If I take these Conditions: $ ( X,||.||) $a normed Vektorspace, $ dim X>1 $ $ \{ 0 \} \neq U \subset X $ closed vector subspace, $…
pastapesto23
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Open ball contains infinitely many disjoint open balls of equal radius

Let $X$ be an infinite dimensional Banach space. Show that the unit ball has contains infinitely many open balls all of equal radius. The hint is to use Riesz's lemma.
user124910
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The Gelfand transformation on $\ell^1(\mathbb Z)$ is not isometric. Do you have an example?

I am looking for an element $(a_n)_{n\in\mathbb Z}\in\ell^1(\mathbb Z)$ with the property $$ \lVert(a_n)\rVert_{\ell^1(\mathbb Z)} > \lVert\sum_{n\in\mathbb Z}a_n z^n\rVert_\infty, $$ where the norm on the right hand side denotes the sup-norm on…
Rasmus
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Why the reversal in inclusion between $l^p$ and $L^p$ spaces?

Anyone who has studied classical functional analysis has been exposed to the sequence spaces ($l^p$ spaces) and the Lebesgue spaces ($L^p$ spaces). It's known that there is a reversal of inclusion in these spaces when the underlying manifold of…
Uche Opara
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Reference request for proof of L.Schwartz's theorem

The following statement is used by Otto Forster (page 224 of Lectures on Riemann Surfaces) in the proof of finiteness of sheaf cohomology group $H^1(Y,\mathcal O_E)$ : Theorem of L. Schwartz. Suppose $E$ and $F$ are Fréchet spaces and $\varphi$,…
Y.H. Chan
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