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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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prove the uniform convergence of $\sum_{n=1}^{\infty}\frac{a_n\sin(nx)}{n}$ and $\sum_{n=1}^{\infty}\frac{b_n\cos(nx)}{n}$

Let $f∈L_2[-\pi,\pi]$ and $a_n,b_n (n=1,2,....)$ be a Fourie coefficients of $f$ in the trigonometric system. Prove the uniform convergence of $\sum_{n=1}^{\infty}\frac{a_n\sin(nx)}{n}$ and $\sum_{n=1}^{\infty}\frac{b_n\cos(nx)}{n}$. Can I just use…
user1223
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Is the kernel in an integral transform considered as some generalized basis?

From Wikipedia An integral transform is any transform $T$ of the following form: $$ (Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt $$ $K: \mathbb{R}^2 \to \mathbb{C}$ is called the kernel function or nucleus of the transform. Is…
Tim
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weak* convergence of Riemann integrals.

Consider the continuous linear functionals $\ell_n, \; n\geq 0$ defined on $C([0,1])$ by $$ \ell_0(f) = \int_0^1 f(t)dt, \;\; \ell_n(f) = \frac{1}{n} \sum_{k = 0}^{n-1} f(\frac{k}{n}), n\geq 1, \;\; f\in C([0,1])$$ Show that $(\ell_n)$ converges…
Johan
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Let $L(A,B)$ bounded operators, $A$ and $B$ Banach . Then $S$ the set of onto operators is open.

I have been trying to prove this fact, but I am having trouble. I cannot even understand it intuitively. What makes it impossible to make the image grow until it is the whole space?
Francisco Maion
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Image of a convex set under a compact map

How do I prove that $T(\overline\Omega)\subset\overline\Omega$, where $$T:\overline\Omega\rightarrow X$$ is compact (X Banach, $\Omega\subset X$ convex and bounded) and $$T(\partial\Omega)\subset\Omega$$ Is this even true for any convex subset of X?
Danut Preda
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Extreme points and Krein-Milman theorem

I am working through the Krein-Milman theorem and using Brezis "Functional Analysis and Sobolev Spaces" problem #1 as an outline. There is one issue that I am struggling with and that is the following: Let $a\in K$, $a$ is an extreme point of $K$…
Joe
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Mazur Theorem implies that if $x_n \rightharpoonup x$ then $|x| \leq \liminf |x_n|$.

In Lax`s Functional Analysis he affirms that the following result due to Mazur: Let $K$ be a closed convex subset of $X$ a normed linear space, and $x_n \rightharpoonup x$, then $x \in K$ implies that if $x_n \rightharpoonup x$ then $|x| \leq…
Francisco Maion
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How do you derive properties of the inverse of the function that satisfies a functional equation?

It looks like there should be a way to do it: $e^{x}$ satisfies $f(x+y)=f(x)f(y)$. Meanwhile its inverse, the natural logarithm, satisfies a similar looking but inverted equation: $f(xy)=f(x)+f(y)$. Surely there must be some way to manipulate or…
John Joe
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Checking continuity of a functional

Let $p \in \mathbb{C}[x]$ and $\mathbb{C}[x]$ is endowed with the norm $\|p\| = sup \{|p(t)|: t \in [0,1]\}$. I need to check if the functional $f(p) = p'(0)$ is continuous or not. I tend to think it's not. To prove it I tried to find some sequence…
Invincible
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Orthogonal Projections and Inclusions in Hilbert Spaces

Let $U,V$ be closed subspaces of the Hilbert space $H$ and $P_U, P_V$ the corresponding orthogonal projections on $U$ and $V$, respectively. I need to show: $$ U\subset V \Leftrightarrow P_U=P_VP_U=P_UP_V.$$ Let $x_0\in H,\, \, $and $U\subset V.$…
user249018
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If $a>1$ then $\|f||=\min \left\{\max\{|f(t)| : t \in [0,1]\}, a \int_{0}^{1} |f(t)|dt \right\}$ is not a norm in $C[0,1]$

Show that if $a>1$ then $$\|f||=\min \left\{\max\{|f(t)| : t \in [0,1]\}, a \int_{0}^{1} |f(t)|dt \right\}$$ is not a norm in $C[0,1]$. Can someone help? My idea is to find some functions $f$ and $g$ in $C[0,1]$ such that $\|f+g\| >…
zariski
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Prove that with the Cauchy Schwarz inequality, if $(f,g)=||f||||g||$ and $g \neq 0$ then $f=cg$ for a scalar $c$.

$\textbf{Prompt:}$ Prove that with the Cauchy Schwarz inequality, if $(f,g)=||f||||g||$ and $g \neq 0$ then $f=cg$ for a scalar $c$. I have an outline for suggestions for the proof, but I don't understand it. Here are the suggestions: Assume…
PBJ
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Exercise: If $\{f(x_n)\}$ is Cauchy $\forall f \in X^\ast$ then $\exists x \in X : x_n \rightarrow x$ weakly

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Excercise: $X$ is a reflexive Banach space and $\{x_n \} \in X$. Prove that if $\{f(x_n)\}$ is Cauchy $\forall f \in X^\ast$ then $\exists x \in…
DoubleTrouble
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Functional analysis, bounded operators

One question, if $X$ and $Y$ are two Banach spaces and $A : X \rightarrow Y$ is a linear injective open operator, then $A$ has to be bounded?
Negreanu
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Linear Operator norm for Banach Steinhaus proof

As a prerequisite for the proof of the Banach-Steinhaus theorem, the following is used: For two normed vector spaces $E,F$ and a continuous linear mapping $T: E\rightarrow F$, for all $x_0\in E,r\ge0$ it holds that $$\sup_{||x-x_0||_E\le r}…
Jan K
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