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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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non separable implies an uncountable set with lower bounded distances?

Given a Banach space the only way i've seen to show that it is not separable is to show that there is a more than countable set $A$ and a costant $c>0$ such that $|a_1-a_2|>c, \forall a_1 \neq a_2 \in A$(in this way you show that $l^{\infty}$ is not…
Jimmy1790
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How to calculate the eigenvalues of an operator?

I am trying to determine the eigenvalues of the following operator. $T:L^2[0,1] \rightarrow L^2[0,1]$, $Tf(x)=\int_0^1(2xy-x-y+1)f(y)dy$ The Eigenvalues are $\rho_p(T):=\{\lambda: \lambda-T \text{ is not injective}\}$ My approach: Let $\lambda \in…
wanymose
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Is the cardinality of the dimension of an infinite-dimensional vector space well-defined?

I thought of this question because of something in physics. In quantum mechanics, the state of a system is associated with a unit vector in a Hilbert space, which in many cases can be thought of as $L^2$. Physicists often project this vector to…
Frank_N1ng
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Intersection of Projections on Hilbert space

Let $H$ be a Hilbert space, on which $P,Q$ be projection operators. Let $S:=\mathcal{R}(P)\cap\mathcal{R}(Q)$ be the intersection of ranges, then it is easy to show the orthogonal complement $S^{\bot}$ is an invariant subspace of $PQP$. The question…
Roy Han
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Functions as integrals of basis functions

I was wondering if there are spaces (function spaces) where the functions have an integral representation, i.e. can be written as an integral involving Fourier coefficients and basis functions, akin to the Fourier transform. If so, what are they…
echoone
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Let $f_n$ be a$f_{n+1}\left(x\right)=\frac{1}{x}\int _0^xf_n\left(t\right)dt\:$

Let be $f_n $ be a sequence of functions and $f_0$ an continuous arbitrary function derivable in $0$ such that: $$f_{n+1}\left(x\right)=\frac{1}{x}\int _0^xf_n\left(t\right)dt$$ for every $n$ positive integer. The domain of $f_n$ is $[0,1]$ I was…
shangq_tou
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Continuity of orthogonal projection in Hilbert space with respect to different inner products

Suppose we are given a Hilbert space A, an infinite dimensional closed subspace B and an inner product $<\cdot,\cdot>$. A sequence of inner products $<\cdot,\cdot>_i$ converge to $<\cdot,\cdot>$ in the sense that $C^{-1}||v||\le||v||_i\le C…
Tian LAN
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Compactness in sequence space

We have the vector space $X=\{\vec{x}=(x_1, x_2,\cdots) | x_n\in\mathbb{R} (n\in\mathbb{N}), \sum_{n=1}^{\infty}\frac{1}{n}|x_n|<\infty \}$, and the norm $\|\vec{x}\|=\sum_{n=1}^{\infty}\frac{1}{n}|x_n|\ (\vec{x}=(x_1, x_2, \cdots))$ on it. This…
hoge
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Basis is uncountable or countable when X is separable?

The following is from Functional Analysis book by Conway: Let $X$ be a separable infinite-dimensional Banach space and let ${\{e_i : i \in I}\}$ be a Hamel basis for $X$ with $|| e_i || = 1$ for all $i$. Note that a Baire Category argument shows…
user200918
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How to show operator is compact

Currently I'm self studying functional analysis, namely compact operators. In the text, the author gives the following example: Example 1: Let $C_1$ and $C_2$ be positive constants and let $$ M:=\left\{x(t)\in C[a,b]:|x(t)|
Rough_Manifolds
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If $A$ is normal show that $A+A^*$ bijective implies $A$ is bijective

Suppose $A$ is a normal ($||Ax||=||A^*x||$ for all $x \in H$ ) bounded linear operator on a Hilbert space $H$. We want to show that if $A+A^*$ is a bijection then so is $A$. A is injective is very easy since $kerA=kerA^*$ we have that if $Ax=0$…
Muselive
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Vector-valued integral: Equivalence of Riemann sums and Rudin's definition

Definition/Notation: In his book "Functional Analysis", Rudin defines the integral of a vector-valued function as follows (Definition 3.26). Let $(Q, \mu)$ be a measure space, $X$ a topological vector space on which the dual $X^*$ separates points,…
Xander
  • 159
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Compact embedding of $l^p$ to $c_0$

if $c_0$ is a space of real sequences that converges to zero with sup norm. I can show that $l^{p} \hookrightarrow c_{0}$ embedding is not compact with the sup norm. But I want something more intresting than that. I want to find norm on $c_0$ such…
a.p
  • 338
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An interesting application of the Hahn-Banach Theorem.

Let $X$ be a normed space and $x_1,x_2\in X$ nonzero elements. Show that there are functionals $F_1,F_2\in X'$ such that $F_1(x_1)F_2(x_2)=\lVert x_1\rVert \lVert x_2\rVert$ and $\lVert F_1\rVert \lVert x_1\rVert =\lVert F_2\rVert \lVert…
Gerschgorin
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Give an example of a continuous linear operator $\displaystyle\|T\|=\sup_{\|x\|\le1} \|T(x)\|$ such that the supremum not reached

Let $T:X\longrightarrow Y$ be a continuous linear operator , $X \;,\;Y$ normed spaces with $$\|T\|=\sup_{\|x\|\le1} \|T(x)\|$$ Give an example of a continuous linear operator such that the supremum not reached $$\|T(x)\|<\|T\|\;\; ,\;\; \|x\|\le…
felipeuni
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