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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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Dual of an isometry

Let $T : X \to Y$ be a linear isometry between normed spaces $X,Y$. Must the dual map $T^* : Y^* \to X^*$ be an isometry?
Doug
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Limited operator

I am trying to prove that the given operator is bounded in the $L_2[0, 1]$ space: $$ (Ax)(t) = t^{r-1} \int\limits^{t}_{0} \frac{x(s)}{s^r} ds $$ The way I'm trying to do it is via a Hölder's inequality, Fubini's theorem and also using Hardy's…
wxist
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Codimension in the completion

Let $X$ be a normed space and let $Y\subset X$ be a subspace of codimension 1. Let $\operatorname{cl}_{\widehat{X}}(Y)$ be the closure of $Y$ in the completion $\widehat{X}$ of $X$. Is still the dimension of…
complete
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Does inverse mapping theorem holds for an incomplete subspace of a Banach space?

Let $l^1$ be the space of all absolutely convergent series, and $$f:l^1\to l^1$$ be a $C^1$ (or $C^\infty$ if it is necessary) mapping satisfying $$f(0)=0$$ $$\nabla f(0) = I$$ then the inverse mapping theorem guarantees that $f$ has a $C^1$ inverse…
qdmj
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Does every absorbing set of a Banach space contain a neighborhood of origin?

Let $X$ be a Banach space and $A$ be any absorbing subset of $X$. Does $A$ contain a neighborhood of the origin?
Shailesh
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Simple proof of Krein-Smulian

I have a question concerning a simple and short proof of Krein-Shmulian, the proof can be found here https://people.math.ethz.ch/~jteichma/slides_ftap.pdf on page 35/36. Here the proof: Let $X$ be a Banach space. The Krein-Smulian theorem tells that…
Linda Weder
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$L^2$-lower semicontinuity of an integral operator on $G(x,\nabla w(x))$

In the paper "A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations" by Nochetto, Savaré, Verdi we find the following claim in Example 2.4: Let $\mathcal H:=L^2(\Omega),\ p>1$ $$ \phi(w):=\int_\Omega…
juisoo
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Prove that $\lim_{n\rightarrow\infty} \langle S_{k_n}u,x\rangle=\langle u,y\rangle$ using Banach-Alaoglu Throrem

Let $\mathscr{H}$ be separable Hilbert space and let $\{S_n\}\subseteq B(\mathscr{H})$ satisfy $\sup_n\|S_n\|_{\mathscr{H}\rightarrow\mathscr{H}}=M<\infty$. Fix $x\in \mathscr{H}$. Prove that there exists $y\in\mathscr{H}$ and subsequence…
user640718
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Why $\mbox{Ker }T^{*}\oplus\overline{\mbox{Im }{T}}=X$?

Can you explain me or indicate where can I find a proof, please why: $$\mbox{Ker }T^{*}\oplus\overline{\mbox{Im }{T}}=X \mbox{ ? }$$ $X$ is a complex Hilbert space. thanks :)
Iuli
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Can you project unto closed subspaces of normed spaces that are not necessarily pre-Hilbert?

I'm working through some notes for my signal processing class and they introduce the whole notion of pre-Hilbert spaces (inner product spaces) essentially only in order to be able to project elements onto closed subspaces. My question is: suppose…
Othman El Hammouchi
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Fredholm operators and Compact operators

Suppose $X$ be an infinite dimensional Banach space. How to prove that: $A$ and $B$ are two Fredholm operator on $X$, if $\mathrm{index}(A)=\mathrm{index}(B)$, then there exists an invertible operator $C$ such that $A-BC$ is compact??? I know…
CQUMath
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Functional Analysis Exercise Problem

Let $X$ be a Banach space and let $A\colon X\to X^*$ be a linear operator satisfying $$(Ax)(x)\geq 0\quad\forall x\in X.$$ Show that $A\in L(X,X^*)$. Here, $L(X,X^*)$ stands for the set of bounded linear transformations from $X$ to $X^*$ (the set…
Rioghasarig
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Is $C^{k,\alpha}(\mathbb R^n)$ reflexive?

Here $C^{k,\alpha}(\mathbb R^n)$ refers to the usual Banach space of functions that are $k$ times continuously differentiable, have bounded derivatives, and whose $k$th derivatives have finite Hoelder norm, with Hoelder exponent…
Arthur Trask
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Proving Positivity of Extension of Linear Functional on $C(X)$

Question I am currently working through Royden's proof that any continuous linear operator $L$ on $C(X)$ can be written as the difference of two positive linear functionals on $C(X)$, where $X$ is a compact Hausdorff space equipped with the maximum…
TotemRobes1889
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Existence of a `Partial Weak Limit' in $L^1$

Suppose that I have a sequence of functions $f_n\in L^1(\mathbb{R}^d)$ for which $\lim_{n\rightarrow \infty} \int_{\mathbb{R}^d} f_n(x)g(x)dx$ exists for all $g\in S\subset L^\infty(\mathbb{R}^d)$, where $S$ is a closed subspace of…
jwsiegel
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