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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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Show that $\{e_n\}_{n≥2}$ is an orthonormal system in which is not complete. However, if $f \in E$ and $f⊥e_n$, for all n≥2, then =0.

Consider $x_0 = (1/j)_{j≥1} \in l^2(\mathbb{N})$ and $\{_\}$ the usual canonical vectors of $ l^2(\mathbb{N})$. Then $E = span(x_0,e_n)_{n≥2}$ is a pre-Hilbert space. Show that $\{e_n\}_{n≥2}$ is an orthonormal system in which is not complete.…
Laura Merchán
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convergence of $e_n$ in $l_1$

Let $X=c_0$ and $X_2=c$ with norm $||x||_{\infty}$. Both $X_1^*$ and $X_2^*$ are isometrically isomorphic to $l_1$. I know that $e_n\to 0$ in weak* topology in $l_1$ if I consider it as dual of $c_0$, but can I make a similar conclusion when I…
Shweta Aggrawal
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Is the unit ball in $X''$ weak-star closed?

Is the unit ball in $X''$ weak-star closed? I reached a point in my argument where it would suffice to show that the unit ball in $X''$ is weak-star closed. (Where $X$ is just some topological space and $X'$ is the continuous dual of $X$, etc.) If…
user58514
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Linearization of operator

Suppose $K:X\to Y$ is nonlinear operator between two Banach spaces. That is $f\mapsto K(f)$. The linearized operator is written as $$Kf_0 + K'f_0(f-f_0)$$ I would like to ask is it correct to say that $K'$ is a Frechet derivative? Frechet derivative…
Bayes
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In a topological vector space, if $A$ is compact and $B$ is closed, then $A+B$ is closed.

If $x\notin A+B$, then there is a neighborhood $V$ of $0$ such that $(x+V)\cap (A+B)=\emptyset$. But I am stuck in this step. I have seen some solutions concerning about $\mathbb R^n$, but I don't know how to apply it to topological vector space…
Knt
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Closed subspace $E$ of $L^p(\Omega),\ E\subset L^{\infty}(\Omega),\ ||f||_{\infty}\le M||f||_2$

Let $p\in [0,\infty)$, $(\Omega,\mathcal{F},\mathbb{P})$ a probability space and $E$ a closed subspace of $L^p(\Omega)$. We suppose that $E\subset L^{\infty}(\Omega)$. How do I prove that $\exists M>0$ such that $||f||_{\infty}\le M||f||_2\ \forall…
Tengen
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Closed graph theorem and equivalence of norms in $C[0,1]$

Consider $id:(C[0,1],||.||_\sup) \mapsto (C[0,1], ||.||_{*}$ Show that up to an equivalence of norms, the supremum norm is the only norm on $C[0,1]$ which makes C[0,1] complete and which also implies the point-wise convergence. Sketch: I consider…
Blabla
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Why is an operator's spectrum closed?

I'm trying to prove that the spectrum of an operator is closed. I was thinking of using the fact that the set of invertible operators is open, and hence the set of non-invertibles operators must be closed. Is it ok? Also, if I wanted to go by a…
An old man in the sea.
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Show that an open mapping need not map closed sets onto closed sets

My question is the following one Question - Show that an open mapping need not map closed sets onto closed sets I have no idea of how to solve this particular problem. Do you have any good suggestions? Thanks in advance!
DonMath
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riesz frechet for a operator

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a unique $g_{x}\in H$ with $\langle f,g_x\rangle=(Tf)(x)$…
Lech121
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Director sum, isometric

let $ E $ be a normed space that can be represented as a direct sum of two vector subspaces: $ E = F + G $. Show that $\frac{ E}{ F}$ is isometric with $ G $ Using the quotient norm since $ F $ is a closed subspace I want to build the bijective…
Kevin
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Determine the adjoint of a linear operator

Let $A:C^2[0,1]\to L^2[0,1]$ and $Af=f''$. Then we know that $A$ is an densely defined unbounded operator. How to determine $A^*$? I know that this operator is not closed since it is not bounded: $||T(\sin(nx))||\geq ||\sin(nx)||$. Furthermore, it…
user715371
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Is $\{\phi \in \mathcal C ([0,1], \mathbb R) \mid \forall k \in\mathbb N^+:\int^1_0 x^k\phi(x) =0\}$ a singleton?

Let $$A=\{\phi \in \mathcal C ([0,1], \mathbb R) \mid \forall k \in\mathbb N^+:\int^1_0 x^k\phi(x) =0\}$$ Clearly, the function $[0,1] \to \mathbb R, \, x \mapsto 0$ belongs to $A$. I would like to ask if $A$ contains any other element. Thank you so…
Akira
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Calculating interest rate

I'm trying to solve the following problem: The interest rate of deposit is 0,30 at the beginning of a year and 0,5 at the end. Find the profit of deposit of 20 000$. Could you help me?
user709660
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Dual space of $\mathcal{C}_0[a,b]$

I want to calculate the dual space of $\mathcal{C}_0[a,b]$, that is the space of continuous functions on $[a,b]$ vanishing at $a$. I know that the dual of $\mathcal{C}[a,b]$ is the space of differences of Lebesgue-Stieltjes measures associated to…
Filippo
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