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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 that $\forall \delta>0,\delta B_Y\subseteq T(B_X)\Leftrightarrow B_Y(Tx_0,\delta) \subseteq T(B_X(x_0,1)) \forall x_0\in X$

Let $X,Y$ normed vectorial space and $T \in B(X,Y)$. Prove that $$\forall\ \delta>0,\delta B_Y\subseteq T(B_X)\iff B_Y(Tx_0,\delta) \subseteq T(B_X(x_0,1))\; \forall\ x_0\in X$$ Here $\delta B_Y=\{y\in…
Giulia B.
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Is PUB $ \Longrightarrow $ CGT still an open problem?

Closed graph theorem says ' A closed operator between Banach spaces is continuous'. Principle of uniform boundedness says ' If $V$ is Banach and $W$ is a normed space, then pointwise bounded family of continuous operators between them is uniformly…
ogirkar
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How can we construct an example of isometric linear operator?

How can we construct an example of isometric linear operator $T: H \rightarrow H$ which is not unitary but maps the Hilbert space $H$ onto a proper closed subspace of $H.$ My attempt I found this proof here Prob. 9, Sec. 3.10 in Kreyszig's…
jasmine
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Internet courses on Functional analysis

Some years ago, Coursera offered a MOOC (Massive Online Open Course) on Functional Analysis: An Introduction to Functional Analysis. Taught by John Cagnol. Ecole Centrale Paris. The duration was 8 weeks: 1 Topology; continuity and convergence of a…
M.González
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Modulus of convexity of the space of continuous functions

Let $(X,\|\cdot\|)$ be a Banach space, and $B_{1}$ its closed unit ball. Recall that the modulus of convexity of $X$ is the function $\delta_{X}:[0,2]\longrightarrow [0,1]$ given by $$ \delta_{X}(\varepsilon):=\inf\{1 -\frac{\|x+y\|}{2}:x,y\in…
user123043
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Exercise 1.2 item 2. in Brezis Functional Analysis: Computing the dual norm and the duality map

Consider the following problem, item 2. of the exercise 1.2 in Brezis's Functional Analysis, Sobolev Spaces and Partial Differential Equations: Let $E$ be a vector space of dimension $n$ and let $(e_i)_{1 \leq i \leq n}$ be a basis of $E$. Given $x…
Danilo Gregorin Afonso
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Density property in Sobolev spaces

I would like to ask if there is any clear (in details) proof of the fact that if the dimension $ N \geq 2 $ and $ 1 < p < + \infty, $ then the space $ C_0^{\infty}( \mathbb{R}^N \setminus{ 0}) $ is dense in the Sobolev space $ W^{1,p}(…
SemiMath
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Prove the linear function $f$ is bounded.

Suppose $f$ is a linear functional $(1)$ defined on $C[a,b]\quad(3)$, and $\forall x\in C[a,b],x(t)\geq 0\Rightarrow f(x)\geq 0 \quad(2)$. Prove that f is continuous. Furthermore, prove that there exists a monotonically increasing function $v(t)$…
Yuan
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Is $L^p$ dense in $L^p+L^{\infty}$?

With $p \ge 1$, and norm $\left| \! \left| f \right| \! \right|_{(L^p+L^{\infty})(\mathbb{R}^n,\mathbb{R})} = \inf_{g \in L^p, h \in L^{\infty}, f =g+h} (\left| \! \left| g \right| \! \right|_{L^p} + \left| \! \left| h \right| \!…
Lulu
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To what extent is the space captured by functionals on it

I want to work in the most general settings. Let us say we have a normed space $\mathcal{B}$. To what extent can we capture properties on the space by studying the functionals on this space?
user761210
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Finding a functional on $l^{\infty}$ with certain properties

My task is to show that there exist a linear functional $\varphi$ on $l^{\infty}$ such that: $\varphi(x) = \lim x_n$ for $x \in C$ ($C$ denotes space of sequences with limits), $\vert \vert\varphi \rvert\rvert = 1$. Let's define…
Hendrra
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Prove this closed set has empty interior

I'm working on the space $\ell_1$ under the $\left \| \cdot \right \|_{\infty}$ and I want to prove the set $$ A=\left \{ (x_n) \in \ell_1: \; \sum_{n=1} ^{\infty} |x_n|\leq 1 \right \} $$ is $(a)$ closed, $(b)$ convex, $(c)$ absorbent, $(d) $…
ipreferpi
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Fourier series and uniform convergence for a given function.

The Fourier series of $x$ is $$\sum_{n=1}^{\infty} 2(-1)^n\frac{\sin(nx)}{n}$$ I can see that the series does not converge pointwise to x in $x=\pi$ My question is: what happen in $[0,a]$ when $a<\pi$ ,my intution says that we should get uniform…
Moshe Levy
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If X ist not Banach are weak* compact sets always norm bounded

A subset of the norm dual of a normed space is weak* compact if and only if it is weak* closed and norm bounded. This is stated without a proof in: Infinite Dimensional Analysis A Hitchhiker's Guide Authors: Aliprantis, Charalambos D., Border, Kim…
H.K.
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Prove there exists a sequence $\{y_n\}\subset (0,\infty)$ such that $\bar{\lim_{n\rightarrow \infty}}y_n = \infty$

Let $\{a_n\}\subset \mathbb{R}$ a sequence of positive numbers such that the serie $\sum_{n=1} a_n $converge Prove exists a sequence $\{y_n\}\subset (0,\infty)$ such that $$\bar{\lim_{n\rightarrow \infty}}y_n = \infty$$ and $$\sum_{n=1}a_n y_n$$…
MaryOsuna
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