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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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Complex functions, integration

Let $L_iL_j-L_jL_i = i\hbar\varepsilon_{ijk}L_k$ where $i,j,k\in\{1,2,3\}$ Let $u$ be any eigenstate of $L_3$. How might one show that $\langle L_1^2 \rangle = \int u^*L_1^2u = \int u^*L_2^2u = \langle L_2^2\rangle$ ? I can show that $\langle…
Tom L
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if $\|x+y\|=\|x\|+\|y\|$, then $\|\alpha x+\beta y\|=\alpha \|x\|+\beta \|y\|$

Let $X$ be a normed linear space. Assume that for $x,y \in X$, we have $||x+y||=||x||+||y||$. Show that $||\alpha x+\beta y||=\alpha ||x||+\beta ||y||$ for every $\alpha,\beta \geq 0$. My attempt: Suppose $\beta \geq \alpha$. Then $\|\alpha x+\beta…
Idonknow
  • 15,643
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Hahn–Banach Theorem for Normed Spaces: not unique extension

Let $\ell^{\infty}$ be the set of bounded sequences in $\mathbb{F}$, with the supremum norm. $c \subset \ell^{\infty}$ the sequences whose limit exists. Then there exists a $f \in (\ell^{\infty})'$, the dual, such that $f(x) = \lim_{n \to \infty}…
MrReese
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Maximal ideals in the algebra of all continuously differentiable complex functions on $[0,1]$

Let $A$ be the algebra of all continuously differentiable complex functions on the interval $[0,1]$ with pointwise multiplication, normed by $$ ||f||=||f||_{\infty} + ||f'||_{\infty}. $$ I have to show that $A$ is a semisimple commutative Banach…
simon
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The dual space as finite dimensional subspace

I've started to think about the dual $(\mathbb{R}^n)^*$ of all linear continuous functionals on $\mathbb{R}^n$ as a subspace of all real valued continous functions on $\mathbb{R}^n$. As I understand it the linear space of all real valued continuous…
harajm
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In closed set in a vector normed space, is every point the limit point of some cauchy sequence?

We're concerned only with a normed vector space here. The book "Introduction to Functional Analysis" by Kreyszig says that if we take the closure $\overline{A}$ of set $A$, then for every $x\in \overline{A}$, there exists a sequence $(x_j)$…
user67803
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Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$

I'm reading a book functional analysis, and reading and have seen an example of somebody please help me if you can. The example that I've seen is the following: If $A$ is a normed algebra and $a,b\in A$ so that $ab=ba$, then $$v(a+b)\leq…
Madrit Zhaku
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missing a condition for convergence or not

In preparation of a test I have some problems solving the following problem: Let $A:=\{a\in \ell^{2}: \phantom{x} \|a\|_{2}\leq 1\}$, $(a_{n})_{n} \in A, a \in A$. Prove that for all $b \in \ell^{2}$ the following two statements are…
PiotrMATH
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Proving that the Fourier coefficients of a functional determine it

Proving that the Fourier coefficients of a functional determine it I have the following exercise, taken from old homework of a functional analysis course: Let $\mu\in C(\mathbb{T})^{*}$. Define the Fourier coefficients of $\mu$ by $$…
Belgi
  • 23,150
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Modulus of Continuity of a $x^{\alpha}$

How do I show that modulus of continuity, $\omega(f,\delta)$ where $f(x):=x^{\alpha}$, is $\leq c\delta^{\alpha}$ where c is some constant independent of $\delta$. ($0 <\alpha < 1$)
mathamateur
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A Min-Max-Theorem for self-adjoint operators

I got a small question concerning the proof of a min-max theorem for selfadjoint operators that I'm currently trying to understand. The article I'm refering to is http://en.wikipedia.org/wiki/Min-max_theorem#Compact_operators Section: Compact…
Stan
  • 31
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$P_n[0,1]$ be the set of all polynomial of degree atmost $n$ with supnorm is it a closed in $C[0,1]$?

$P_n[0,1]$ be the set of all polynomial of degree atmost $n$ with supnorm is it a closed in $C[0,1]$? and $P[0,1]$ is set of all polynomials in $C[0,1]$ I know which is dense in $C[0,1]$ as we know by Weirstrass Polynomial approximation theorem any…
Myshkin
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Convex set with convex complement has non empty interior?

My question is if in a normed space (maybe complete if necessary), a convex (non empty) set with convex complement has non-empty interior. I cannot think of a counterexample, but neither how to prove it. Any ideas?
Eparoh
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Norm, convexity and composition of functions

Claim 1: {convex and non-decreasing func.} ◦ {convex func.} is convex Claim 2: Norms are convex (The triangle inequality is the same thing as convexity of the norm) Claim 3: {convex and non-decreasing func.} ◦ {metric} is probably not a metric…
Il Prete Rosso
  • 323
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Continuous function with respect to norm

Let $P_2$ be the space of all polynomial of degree no more than 2 considered as functions on the interval $[0,2]$. For $p$ and $q$ in $P_2$, let $$\langle p,q\rangle=\int_0^2P(x) \overline {q(x)} \, dx$$ and let $\|\bullet\|_2$ be the norm…
User69127
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