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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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Closed subspace consisting of continuous functions in $L^{2}([0,1])$ is finite-dimensional

I wish to show that any closed subspace $M$ of $L^{2}[0,1]$ consisting of continuous real-valued functions is finite-dimensional (which amazes me quite a bit, I must admit). My approach proceeds as follows: Step 0. Note that $L^{2}[0,1]$ is a…
ferhenk
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Finding the Spectrum of integral operator

I have the following integral operator: $$(Ku)(x)=\int_{0}^{1} k(x,y)u(y) \mathop{dy}$$ with $k(x,y)=$min $ \{ x,y \}$ for $0 \leq x,y \leq 1$.$\\$ I have already shown $K$ is a compact, self adjoint operator but now I want to find the spectrum of…
George
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Can a subspace have a larger dual?

I cant manage to figure this out, for instance $L^{1}[0,1]$ has $L^{\infty}[0,1]$ as dual and $C[0,1]$ (a sub space of $L^{1}[0,1]$ have the signed measures of bdd. varitaion as dual. I cant mange to prove anything regarding cardinality realtions…
user123124
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gradient flow and what is, for example, $L^2$ gradient?

Am I right that the gradient flow of a functional $E$ is $$f_t = -\nabla E(f).$$ Solving this for $f$ gives you a minimiser of $E$ in some way? Here the $\nabla$ denotes the gradient or the first variation or Gateaux derivative or whatever is…
Court
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Conditions for Linear Independence for functions defined by integration

Given that the set of strictly positive and continuous functions $$f_i(x,y) >0, \quad i=1,\dots,n$$ are defined on $[0,1]^2$ and $\mathbb{R}$-linearly independent for $(x,y) \in [0,1]^2$. That is if $c_1, \ldots, c_n \in \mathbb{R}$ and …
confused_dragon
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A Banach Space cannot have a denumerable basis:Why is it true?

I came across the following theorem: A Banach Space cannot have a denumerable basis which has been proven in my book. I can't understand why is it true since $\mathbb R$ is a banach space over $\mathbb R$ and it has a countable basis i.e…
Learnmore
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weak* continuous linear functional is in the predual

Let $X$ be a Banach space and $X^*$ its dual. We know that the weak* topology is the least topology that makes every $x \in X$ continuous as an evaluation functional. However, this does not imply that every weak* continuous linear functional is…
Nick Kolliopoulos
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Orthonormal Sets in Hilbert Spaces

let $H$ be a Hilbert space and let M be a dense linear subspace of H. Can we find a complete orthonormal set $\{u_{\alpha}: \alpha \in A\}$ for H in M? I think the answer is negative in general, but I cannot find a counterexample. Thank you very…
Maurizio Barbato
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The the image of the unit ball in X is weak-* dense in the unit ball of X**

I am trying to prove the following theorem and am stuck. Let $X$ be a Banach space. The the image of the unit ball in $X$ is weak-* dense in the unit ball of $X^{**}$. My proof idea Assume $X^{**}$ is equipped with its weak-* topology. Let $Q$ be…
img_teacher
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The adjoint of finite rank operator is finite rank

If $T \in \mathfrak{B}_{00}(\mathfrak{H},\mathfrak{K})$, show that $T^{*} \in \mathfrak{B}_{00}(\mathfrak{K},\mathfrak{H})$ and $dim(ran T) = dim(ran T^{*})$. The $\mathfrak{B}_{00}(\mathfrak{H},\mathfrak{K})$ is the set of continuous finite rank…
Mobius
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Properties of Hardy operator $T(u)(x)=\frac{1}{x}\int_0^x u(t)dt$

Let $u$ be a measurable function in $[0,1]$ and define $T:L^p(0,1)\to L^p(0,1)$ by $Tu(x)=\frac{1}{x}\int_0^x u(t)dt\quad\forall x \in [0,1]$. Let $1
user62138
  • 1,167
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Understanding positive definite kernel

From Mercer's Theorem: A kernel is a symmetric continuous function $ K: [a,b] \times [a,b] \rightarrow \mathbb{R}$, so that $K(x, s) = K(s, x)$ ($\forall s,x \in [a,b]$). $K$ is said to be non-negative definite (or positive semi-definite) if and…
Tim
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Dual of "Dual of Fréchet Space with Weak*-Topology" Equals Dual of "Dual of Fréchet Space with Topology of Compact Convergence"

Let $X$ be a Fréchet space. I know that: Closed convex, balanced, hulls of compact subsets of $X$ are compact. Let ${X^*}$ denote the (topological) dual space. I know that: ${X^*}$ is also the dual of "$X$ with the weak topology (induced by…
Wayne
  • 611
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Is the weak-star topology on the dual of a Banach space completely regular?

Does the weak-star topology on the dual of a separable Banach space make the dual completely regular under weak-star topology? So I have come to the stage in a proof where if I could show this, then I would be done! In case you are interested in the…
user58514
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Does the Gelfand transformation on $\ell^1(\mathbb Z)$ possess a continuous inverse on its image?

I am interested in the Gelfand transformation $$ \Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n. $$ This is an injective homomorphism of Banach algebras. It is neither isometric nor surjective. However,…
Rasmus
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