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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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The space C(X) and uniformly convergence on compact subsets of X

Let $X$ be a completely regular space. For every compact subset $K$ of X, define a seminorm $p_K: C(X)\to {\Bbb C}$ such that $p_K(f):=\sup_{x\in K}|f(x)|$. Then $\{p_K;K ~is ~compact \}$ is a collection of seminorms that makes $C(X)$ a locally…
user 1253
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Norm of Volterra Operator

Here is a structured question of my assignment: Given the operator $K:C[0,1]\mapsto C[0,1]$ defined by$$(Kf)(t)=\int_{0}^{1}k(t,s)f(s)ds.$$ (1) Prove that $K$ is bounded linear operator and $$\left \| K \right \|\leq \max_{t\in…
nam
  • 733
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Extending a functional with same norm

I have: $X = <\mathbb R^2, ||(x,y)|| = \sqrt{4x^2+y^2}>, L = \{(2x,3x), x \in\mathbb R\}, $ $\phi_0 \in L^* : \phi_0 (2x,3x) = -2x$ I need to extend $\phi_0$ on $X$ without changing norm (norm should stay the same). How can I do it?
michaeluskov
  • 73
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Lipschitz function and uniform boundedness principle

Let $(S,d)$ be a metric space and $X$ be a normed space. Show that if $f:S\to X$ is a function such that for all $L\in X^*$, $L\circ f:S\to {\Bbb F}$ is lipschitz(there is a constant $M>0$ such that for all $s,t\in S$, $||f(t)-f(s)||\leq Md(s,t)$ )…
nika
  • 171
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Prove that weak convergence does not necessarily imply strong convergence without counterexample.

Here is the set of original problems. Let $\{x_n\}$ be a sequence in a normed linear space $X$. Prove that: Strong convergence implies weak convergence with the same limit. The converse of 1. is not generally true. If $\mathrm{dim}(X) < \infty$,…
NasuSama
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A weakly closed subset of $L^p$

I have $\Omega \subseteq \mathbb{R}^n$ such that $m(\Omega) < \infty$ (Lebesgue measure) and $1 \le p < \infty$, and I want to prove that if $C \subseteq L^p (\Omega)$ is closed in the weak topology $\sigma(L^p, L^q)$ of $L^p$, with $1/p + 1/q =1$,…
gangrene
  • 1,517
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the principle of uniform boundedness

If $\{x_n\} \subset \ell^1$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in c_0$ iff $\sup_n||x_n||_1< \infty$ and $x_n(j)\to 0$ for $j\geq 1$. I can proof it by the principle of uniform boundedness; but my question is "why is not it…
nika
  • 171
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Does integration over continuous compactly supported functions completely determine the measure?

If $X$ is a locally compact space. And $\mu, \nu$ are two Radon measures on $X$. If for any continuous compactly supported function $f \in C_c ( X \to \mathbb R )$ we have $ \int f d\mu = \int f d\nu $, does this implies $ \mu = \nu $? This…
user112564
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Show that this linear operator is not continuous.

This is a question of my last exam in Functional Analysis of my graduation: Consider that norm: $|f|=\int_0^1 x^2 f(x)$ where $f\in C^0([0,1])$. Show that the linear operator $f(1-x)\in C^0([0,1])$ is not continuous. MY ATTEMPT I show that this…
Felipe
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Prove that the space of continuous linear functionals B(X,Y) is complete iff Y is complete

Let $x$ and $Y$ be normed vector spaces and assume that $X\ne\{0\}$. Prove that $B(X,Y)$ - space of continuous linear functionals $A:X\rightarrow Y$ - is complete with respect to the norm $\|A\|:=\sup_{\|x\|\le1}{\|Ax\|}$ iff $Y$ is complete. My…
luka5z
  • 6,359
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sub-algebra of continuous real valued functions without unit must vanish at a point

If $X$ is compact Hausdorff and $A$ a closed subalgebra ( a vector space and closed under multiplication) of $C( X \to \mathbb R)$ the set of continuous real valued functions which separates points, and $A$ does not contain the unit. How can I prove…
user112564
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the sum of closed convex sets

Let C and D be two closed convex subsets of a Banach space with C+D is closed. If bounded sequence $\{x_n\}\subset C+D$, can we choose bounded sequences $\{c_n\}\subset C$ and $\{d_n\}\subset D$ such that $x_n=c_n+d_n$ for every n.
luozhenghua
  • 134
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First resolvent equation

$(B,||\cdot||)$: banach space A family $(G_{\alpha})_{\alpha>0}$ of linear operators on $B$ with $D(G_{\alpha})=B$ for all $\alpha>0$ is called a strongly continuous contraction resolvent if (i) $\lim_{\alpha\to \infty}\alpha G_{\alpha}u=u$ for…
ko4
  • 541
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Limit of sequence of functionals

I have the following problem: $X$ real Banach space $(\epsilon_n)_{n\geq1}$ a positive sequence converging to zero $(f_n)_{n\geq1}$ a sequence in $X^*$ the dual space of $X$ with the following property: $\exists r>0$ st $\forall x\in B_r(0) \exists…
sky90
  • 1,518
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Strict convexity of $c_0$

Let $c_0$ be a spaces of sequences converging to $0$ with the following norm $$ \|x\|=\sup\{|x_i|: i\in \mathbb{N}\}+\left(\sum_{i=1}^{\infty}\left(\frac{x_i}{i}\right)^2\right)^{\frac{1}{2}} $$ Prove that $(c_0,\|.\|)$ is strictly convex but not…
impartialmale
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