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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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Kantorovich's theorem (a version of Hahn Banach's theorem for positive linear functionals)

I am having some trouble to complete the proof of the following theorem: Consider the set of continuous functions from a compact topological space $X$ into $\mathbb{R}$, namely $C(X,\mathbb{R})$, equipped with the supremum norm. And let…
Modesto Rosado
  • 445
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Does it exist a linear functional?

There is a problem about which I have no ideas at all: For each $t\in [0,1]$, we define the functional $\delta_t$ on $C [0,1]$ as $\delta_t(f)=f(t)$. Let $I (f) = \displaystyle\int\limits_0^1f (t) dt$. Is there a bounded linear functional $F$ on $C…
thing
  • 1,690
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Convex cone and compact subset

I need a little help in the following question. Let $V$ be a real normed vector space and let $K\subset V$ be a compact and convex subset. Let $C\subset V^\prime$ be a convex cone. Assume that, for each $f\in C$, there exists $v_f\in K$ such that…
Mathecm
  • 689
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A variation of the Holder's inequality

Can someone please give me an idea how to go about solving this problem? If $f\in L^p(\Omega)$ show that $$\| f\|_p = \sup\left| \int_\Omega fg dx\right| = \sup \int_\Omega |fg|dx $$ where the supremum is taken over all $g \in L^q(\Omega)$ such…
Teodorism
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Topology on bounded measurable functions generating countably additive measures as dual

Let $X$ be a set, $\mathcal{A}$ an algebra on $X$. Denote by $B(X, \mathcal{A})$ the Banach space of bounded $\mathcal{A}$-measurable functions with the supremum norm and by $ba(\mathcal{A})$ the Banach space of finitely additive signed measures…
yada
  • 3,535
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Left regular action of $L^1(G)$ on $L^2(G)$ is given by convolution.

This seems basic and I couldn't find it by googling so I thought I'd post it along with an answer. If $G$ is a locally compact topological group and $\pi:G\rightarrow U(H)$ is a unitary representation of $G$ on a Hilbert space $H$, we can talk about…
Tim kinsella
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Showing that the unit ball in $(C^\alpha[a, b], \|\cdot\|_\alpha)$ is compact in $(C[a, b], \|\cdot\|_\infty)$

Let $(C[a, b], \|\cdot\|_\infty)$ be the usual Banach space of continuous functions on $[a, b]$ and for $\alpha\in(0,1]$ and $f\in C[a, b]$ define $$ [f]_\alpha = \sup_{x,y\in[a,b];x\neq y}\frac{|f(x)-f(y)|}{|x - y|^\alpha} $$ Let $C^\alpha[a, b]$…
user3002473
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Two Continuous Extensions of an Operator

Let $T: V \to U$ be a linear operator, where $V \subset W$, and suppose there exists two norms on $W$, denoted $\| \cdot \|_1$ and $\| \cdot \|_2$, such that $V$ is a dense subspace of $W$ with respect to either norm. If $T$ is continuous with…
Jacob Denson
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Open mapping $\iff$ bounded inverse $\iff$ closed graph

I've studied closed graph theorem. In particular I first saw the open mapping theorem, then the bounded inverse theorem and finally the closed graph. I saw the proof that if a linear operator is closed then is bounded. Then my notes state ($X,Y$…
Peanojr
  • 312
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Banach spaces and weak topology

Claim : A Banach space V is reflexive iff its unit ball B is weakly compact. So I want to show that 'unit ball is weakly compact => V is reflexive' without using goldstine. So I'm trying to use the theorem of bipolars. From the theorem of bipolars…
user58514
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containment in a cantor set

I said that the following statement was false: If a belongs to the Cantor set and I is an open interval containing a, then I contains at least one other member of the Cantor set. Unfortunately I am unable to know for sure if I am right or wrong as I…
Omar
  • 517
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Commutant of $L^{\infty}(X)$ inside $\mathcal{B}(L^2(X))$

Represent $L^{\infty}(X)$ in $\mathcal{B}(L^{2}(X)$ by the multiplication operator given by $f \to M_f$ where $M_f(g)=fg$. I want to prove that the commutant of $L^{\infty}(X)$ when regarded as a subalgebra of $\mathcal{B}(L^{2}(X)$ is…
budi
  • 1,780
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Let $(x_n)$ be a sequence in a Banach space $X$. Which of the following conditions ensure(s) that $(x_n)$ is convergent in $X$?

Let $(x_n)$ be a sequence in a Banach space $X$. Which of the following conditions ensure(s) that $(x_n)$ is convergent in $X$? (a) $\|x_n − x_{n+1}\| \to 0.$ (b) $\sum^{\infty}_{n=1} \|x_n − x_{n+1}\|<\infty$ I think that the first condition is…
GraduateStudent
  • 1,332
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study inside a finite-dimensional inner product space.

I am studying for my test and have completed most of the problems assigned to me, but am having trouble with a few, and this one in particular. Thank you in advance for any help. Let $V$ be a finite-dimensional inner product space. Suppose that $U$…
Omar
  • 517
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CARLA (Continuous Action Reinforcement Learning Automaton) convergence proof

I am going through this paper on the proof (or better conditions) for convergence of CARLA system by Rodriguez et al (2011). Since I do not come from a pure mathematical background, I need some assistance in understanding parts of the proof or some…
Zero
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