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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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if S and T are Hilbert-Schmidt between two Hilbert spaces U and V, is $\operatorname{trace}_U(S^{\ast}T) = \operatorname{trace}_V(T^{\ast}S)$?

Let $U$ and $V$ be two real Hilbert spaces and $S$, $T$ two Hilbert-Schmidt operators between $U$ and $V$. Denote their adjoints by $S^{\ast}$ and $T^{\ast}$, both viewed after identification of $U$ and $V$ with their respective duals as…
n_flanders
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Surjective bounded linear map bound

Let $X,Y$ be Banach spaces. Let $T: X \rightarrow Y$ be a surjective bounded linear map. Show that there is a constant $M>0$ such that for each $y\in Y $ there is a solution to $Tx=y$ with $\| x\| \leq M \| y\|$. Let $B_Y(0,r)= \{ y \in Y : \| y \|…
Toasted_Brain
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Is the Schwartz Kernel of the Identity an $L^2$ function?

The answer must be no, otherwise the identity would be a H.S. operator. However, whats wrong with the following: We have by Schwartz kernel theorem $k_I( \phi\otimes\psi)=(I\phi,\psi)_{L^2}$ Therefore, for $\psi,\phi \in D(x)$ we have $|k_I(…
yess
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Are polynomials dense in $C^k\left(\bar{B}\right)$?

Let $\bar{B}$ be the closed unit ball in $\mathbb{R}^n$, $C^k\left(\bar{B}\right)$ the Banach space of all real function defined on $\bar{B}$ with continuous derivatives up to order $k$, with norm $$\Vert f \Vert = \sum_{h\le k} \Vert…
AlbertH
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there is no bounded linear functional on $ H$

let $ H= L^2[0,1]$ and $ C^1 $ be the set of all continuouse functions on $ [0,1] $ that have continuouse derivative.Let $ t \in [0,1] $ and define $ L: C^1 \longrightarrow F $ by $ L (h)= h'(t) $. show that there is no bounded linear functional on…
nim
  • 181
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Question regarding continuity of linear operator between banach spaces

Let $T:X \to Y$ linear, $(X,||\cdot||_X), (Y, ||\cdot||_Y)$ being banach spaces. Is it still wrong that $\ker T$ closed $\Rightarrow T$ is continuous? Supposing $(X,|| \cdot||_X)$ is not a banach space, I can think of $$id: (C([0,1]),|| \cdot||_1)…
goalgetter666
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intuition on the isomorphism between $(c_0)^*$ and $(c)^*$

There are a lot of questions on this topic see for example Show that $(c_{0})'$ and $(c)'$ are isometrically isomorphic. . So I´m not interested in a proof rather I wonder how to think about the result. If I understand it correctly both $c_0^*$ and…
Erik
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Why smooth function space is separable under the Holder norm?

Let $C^\infty(R^d)$ be the space of infinite order differentiable function space. As well known that $C^\infty(R^d)$ is not separable under uniform norm $$ \|f\|=\sup_{x\in R^d}|f(x)|. $$ However, it seems that $C^\infty(R^d)$ is separable under…
alphabeta
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Orthogonal complement in $ L_2 $

Find the orthogonal complement of a subspace $$ M = \{ x \in L_2(-1, 1):x(t)=-x(-t), \int_0^1 x(t)t^2dt=0 \} $$ in $L_2(-1, 1).$ As I understand M can be described as all odd functions which are orthogonal to $ \lambda t^2 $ on $(0, 1)$. But I don`t…
s909
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closedness of a subset of a bipolar set

I do not have a very strong knowledge of bipolar sets and all this stuff. Thus it could be that the question is rather easy. However I was not able to prove by myself the closedness of the following set: We are looking at the space…
math
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Fundamental lemma of variational calculus for Bochner space

So I was wondering if there is a "nice" way to prove a variant of the fundamental lemma of variational calc. The setting is as follows: Let $n \in \mathbb{N}$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain with $C^1-$boundary and $T>0$. Let …
kade
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Norm of adjoint proof check

Let ${\cal H},{\cal K}$ be Hilbert spaces. Let $T\in B({\cal H},{\cal K})$. If ${\cal H}={\cal K}$, I know that $||T||=||T^*||$. So I was wondering if the same is true when ${\cal H}$ is not necessarily ${\cal K}$. attempt of proof If $T=0$, then…
chhro
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Is this operator bounded in $C^1[0,1]$?

Let $Ax(t)=x(\sqrt{t})$ in $C^1[0,1]$ with norm $\|x\|_1=\max\limits_{t\in[0,1]}|x(t)|+\max\limits_{t\in[0,1]}|x'(t)|$. It is required to check it for boundedness. All I know is that the operator is not defined on the whole space, but is defined…
thing
  • 1,690
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Continuous linear extension of a multiplication map

Let $S$ denote the unit circle, so that $2\pi$-periodic functions can be treated as functions on $S$. For a given function $g\in C^1(S)$, define the multiplication map $M_g:C^1(S)\to C^1(S)$ by $(M_gf)(x)=g(x)f(x)$. Prove that $M_g$ extends to a…
Kimberly
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Sequence of bounded operators

Here is a problem I'm working on: Suppose $A$ is a compact, Hermitian operator with $\ker A=\{0\}$. Prove that there is a sequence $(B_n)$ of bounded operators so that $$ AB_nx\to x\quad\text{and}\quad B_nAx\to x. $$ Moreover, is it possible to…
jbeard
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