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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 the Friedrichs' mollifier has an integral equal to 1

Prove that the following: $$\int_{\mathbb{R}^{n}}\eta_{\epsilon}(x)dx=1~\forall\epsilon>0$$ Where $\eta_{\epsilon}(x)$ is the Friedrichs' mollifier: $$\eta_{\epsilon}(x)= \frac{1}{\epsilon^n}\eta(\frac{x}{\epsilon}) =\left\{\begin{matrix} …
AdrinMI49
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In normed linear space $X$, if $K$ is convex then $\{y\in X\colon||y-x||=d(x,K)\}$ is convex

As in the title, I'm trying to show that in a normed linear space $X$, if $K$ is convex then $\mathcal{D}_K(x)=\{y\in X\colon||y-x||=d(x,K)\}$ is convex. My attempt : for $\alpha,\beta\in\mathcal{D}_K(x)$ and…
Jimmy
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Are the subsets of homeomorphic spaces also homeomorphic?

Let $W$ be a subset of an $n$-dimensional complex topology vector space $Y$ such that $0\notin W$.We have known that $Y$ is homeomorphic to $C^n$ and let $S$ be the unit sphere of $C^n$.Can anyone show me that $W$ is homeomorphic to $S$?
mathon
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Can we recover the following from, for example, the uniform boundedness principle

It seems to be easy, though I cannot spot the trick. I feel it should use Banach-Steinhaus. Given a normed space $X$, Banach space $Y$ and a sequence of bounded linear maps $T_n\colon X\to Y$ having uniformly bounded norms, suppose that $W\subset X$…
gely
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General case of Arzela Ascoli Theorem.

Consider Exercise 1.1.15 in Salamon’s Functional Analysis, we have: Let $X$ be a compact topological space and let $Y$ be a metric space. A set $\mathscr{F}\subset C(X,Y)$ is precompact iff it is equi-continuous and pointwise precompact. One…
WakeUp-X.Liu
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Spectrum of operator on $\ell^1$ with upper block triangular representation

Any bounded linear operator $T : \ell^1 \to \ell^1$ can be represented by an infinite matrix $A = (A_{ij})$ with $A_{ij} = e_i^T A e_j = (T e_j)_i$. Assume that there is a decomposition of the index set $\mathbb{N} = \bigcup_{i \in K} I_i$ where $K…
yada
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If $f_n \rightarrow f_1$ a.e. and $f_n \rightarrow f_2$ weakly,then $f_1=f_2$ a.e.

It's an exercise 14 of chapter 1 from stein's functional analysis,here the sequence $\{f_n\}$ satisfy $\|f_n\|_{L^p}\leq M<\infty$ and $1
math
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Showing that Fourier transform on $L^2(\mathbb{R})$ is a unitary operator

Definition: An operator $U\colon H\to H$ is called unitary if $\langle Ux,Uy \rangle=\langle x,y \rangle$ for all $x,y\in H$ and $\text{Im}U=H$. Currently I'm self studying functional analysis, namely unitary operators. In the text, the author gives…
Rough_Manifolds
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Proving $|f(x)|$ $\le \| x\|\| f\|$ , where f is a continuous linear functional

I am self studying Functional Analysis and came across this proposition. Suppose, $f$ is a continuous linear functional defined on a Normed Linear space $V$, then $$|f(x)| \le \| x\|\| f\|$$ I don't get how this inequality is derived, I know $f : V…
night_crawler
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Two propositions about weak* convergence and (weak) convergence

Let $E$ be a normed space. We have the usual definitions: 1) $f, f_n \in E^*$, $n \in \mathbb{N}$, then $$f_n \xrightarrow{w^*} f :<=> \forall x \in E: f_n(x) \rightarrow f(x)$$ and in this case we say that $(f_n)$ is $weak^*$-$convergent$ to…
Amarus
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Exercise 5 page 36 in Functional Analysis book of Conway

The following is Exercise 5 page 36 in Functional Analysis book of Conway: Find the adjoint of a diagonal operator (Exercise 1.8). The aforementioned exercise 1.8 read: Exercise 1.8. Let $\{e_n\}_{n\in \mathbb{N}}$ be the usual basis for $l^2$…
user200918
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Eigenvectors of Kernel Operator

Let $K\in L^2([a,b]\times[a,b])$, $K(s,t)=\overline K(t,s)$, define $Tf(s)=\int_a^bK(s,t)\bar f(t)\,dt$ for $f \in L^2([a,b])$ I need to show that the eigenfunctions of $T$ are an orthonormal basis for $L_2([a,b])$ I tried to show $T$ is…
Benuci
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Proving that norm of operator equals $\|\varphi_0\|\|x_0\|$

Suppose $X$ is a Banach space over the field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and let $X'$ denote its dual. Fix $x_0 \in X$ and $\varphi_0 \in X'$. I want to prove that norm of linear map $T:B(X)\rightarrow\mathbb{F}$ given by $$T: A…
Louis
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Map between Bochner spaces induced by map between Hilbert spaces

Assume I have a map $F:U \to V$ where $U$ and $V$ are Hilbert spaces. Consider now the spaces $L^p(T;U)$ and $L^p(T;V)$, where $T$ is a measure space. Will $F$ induce a mapping between these two Bochner spaces? It feels reasonable that it in some…
ejk
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A Basic Question about Interpreting Closed Graph Theorem

The closed graph theorem on wikipedia is given as follows: I'm not understanding how this isn't a tautology. Isn't the definition of continuity that $\{x_n\} \rightarrow x \implies \{f(x_n)\} \rightarrow f(x)$?
yoshi
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