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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 .

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On the properties of the Sobolev Spaces $H^s$

Let $H^s(\mathbb{R}^d):= \{ u \in \mathcal{S}' : (1+|\xi|^2)^{s/2}\hat{u}(\xi) \in L^2(\mathbb{R}^d)\}$. It can be shown that this space is a Hilbert space and that $H^s \subset H^t$ if $t \leq s$. Now suppose we have $t > s>0$ such that $H^s…
user1736
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Exercise: Application of The Uniform Boundedness Principle

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Exercise: Let $1 \leq p,q \leq \infty$ be conjugate exponents. Let $a=(a_1,a_2,...)$ be a sequence such that $\sum_1^\infty a_n x_n$ converges for…
DoubleTrouble
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How is it that the derivative operator is a closed linear operator?

By definition, if the derivative operator $D:C^1[-1,1]\to C^1[-1,1]$ is closed, then it should be the case that, given any sequence $\{x_n\}$ in $C^1[-1,1]$, and given that $x_n\to x$ as $n\to\infty$ for some $x\in C[-1,1]$, we should conclude that…
xzhu
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How can I calculate the weak derivative of $\frac{1}{\sqrt{x}}$?

Define a function $f(x)$ on $\mathbb{R}$ as: $$f(x) =\begin{cases}0&\text{for }x \le 0\\\\\frac{1}{\sqrt{x}}&\text{for }x > 0\;.\end{cases}$$ Then, how can I calculate the weak derivative(of course, distribution sense) of it? I roughly guess in…
Shorah
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A subspace of a dual space is norm closed if and only if it is weak star closed.

I am trying to figure out if the statement holds true, the literature i am following says that its not true but i don't seem to understand, If $Y$ is a Banach space and let subspace $A \subset Y'$, such that $Y'$ is a dual . $A$ is norm closed if…
Theorem
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How to calculate values of the Hahn-Banach-extended integral functional?

Let $\Phi: C[0,1]\longrightarrow\mathbb{R}$ the linear functional defined by $\Phi(f)=\int_0^1 f(x)dx$. Let $\tilde{\Phi}$ an extension of $\Phi$ to the normed space $(B[0,1]$ (of bounded functions on $[0,1]$, with the $\sup$ norm) such that…
alejopelaez
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Is the open mapping theorem true for all complete metrizable topological vector spaces?

Recall that the open mapping theorem is true for Banach spaces. Furthermore, it is also true for Frechet spaces, which are complete metric spaces(with extra properties). It makes me wonder whether we can say that open mapping theorem applies to all…
Keith
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Hahn Banach theorem and sublinear functionals

I find it strange that it is such a useful property for a linear functional to be bounded by a sublinear functional. What information does it really give? Let $X$ be a Banach space and $p,f:X\to\mathbb{R}$. Well I realize that if $p$ is a sublinear…
rom
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How is it useful to know what the dual of a space is isomorphic to?

Wikipedia has a neat spaces and their duals. For example, it lists that the dual of $c_0$ is $l_1$. But how can I use that knowledge? I'm trying to prove something for all functions of the dual of $c_0$, and figured that I could use my familiarity…
Anders
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Representation of linear functionals

I have always seen the linear functionals in $R^n$ expressed at $\ell(x) = \sum_{i=0}^n a_ix_i$ And in an countable metric space $\ell(x) = \sum_{i=0}^{\infty} a_ix_i$. I guess that this follows directly from…
Johan
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Do $e_1$, $e_2$, ... generate the entire $l_2$ space?

From our textbook goes some statement like this: ... let $X$ be the linear subspace of $l_2$ generated by the vectors $$\left\{e_1,e_2,e_3,...\right\}$$ ... Which feels strange to me because I thought $e_1=(1,0,0,...)$, $e_2=(0,1,0,...)$, ... are…
xzhu
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Linear extension and Hahn Banach Theorem. Am I missing some detail in this exercise?

This is an exercise problem from a course in functional analysis. However, it is not a homework problem. I think I got it figured out, however my teacher said something during the lecture that I didn't understand, and was lagging behind with taking…
DoubleTrouble
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Kreyszig 2.10.4 Implication of operator convergence

Problem 2.10.4 from Kreyszig Let $X$ and $Y$ be normed spaces and $T_n:X\to Y (n=1, 2,...)$ bounded linear operators. Show that convergence $T_n \to T$ implies that for every $\epsilon>0$ there is an $N$ such that for all $n>N$ and all $x$ in any…
Konstantin
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Weak derivative of a Lipschitz function

Let $f \in C^\infty_c(\mathbb{R})^*$ be a distribution. How can I show the following: $$f \in C^{0,1}(\mathbb{R}) \Leftrightarrow f \in L^\infty(\mathbb{R}) \text{ and } f' \in L^\infty(\mathbb{R}) \text{.}$$ Here $C^{0,1}(\mathbb{R})$ is the space…
Cantor
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Lax-Milgram theorem on Banach space

Lax-Milgram theorem states that If $B(,)$ is a symmetric,strictly positive and bounded bilinear form on Hilbert space $V$, then for any continuous functional $l$, there exists $u\in V$ s.t. $B(u,v)=l(v)$. I am wondering if this result can be…
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