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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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Hilbert transform on a finite domain

While doing physics research, I`ve come across the linear operator $$(K\phi)(t_{2}):=\text{P.V}\int_{-1}^{1}\frac{\phi(t_{1})}{(t_{1}-t_{2})}dt_{1}$$ with arguments and values in functions on $[-1,1]$. The functional of my interest is extremized by…
Austris Akmentiņš
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Spectrum of a right shift operator.

I have some doubts on the following problem : Let us consider $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $by $(x_1,x_2..... ) \to (x_2, x_3 ........) $. I want to find the eigen values and spectrum of T and also of $T' : \ell^\infty (\mathbb…
Theorem
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Is $M(X)$ (regular borel measures), the dual of $C_0(X)$ separable?

For $X$ locally compact (let's take $X=\mathbb{R}^d$), we know that the dual of $C_0(X)$ is $M(X)$, the space of regular borel measures on X. $C_0(X)$ is separable but is $M(X)$ separable? I have tried searching but haven't seen the result nowhere,…
user44670
  • 435
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Simultaneous orthogonal basis for $L^2$, $H^1_0$, ... $H^k_0$

Let $\Omega \in R^n$ be open, bounded and with smooth boundary. Can you prove the existence of a system of vectors that simultaneously forms an orthogonal basis both in $L^2(\Omega)$ and $H^1_0(\Omega)$? Can you generalize this construction to…
shuhalo
  • 7,485
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Boundedness of an integral operator

Let $K_n \in L^1([0,1]), n \geq 1$ and define a linear map $T$ from $L^\infty([0,1]) $to sequences by $$ Tf = (x_n), \;\; x_n =\int_0^1 K_n(x)f(x)dx$$ Show that $T$ is a bounded linear operator from $L^\infty([0,1]) $to $\ell^\infty$…
Johan
  • 4,002
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The space $C_b(\mathbb{R})$ is complete

Let $C_b(\mathbb{R})$be the space of all bounded continuous functions on $\mathbb{R}$, normed with $$\|f\|= \sup_{x\in \mathbb{R}}|f(x)|$$ Show that this space is complete. Complete mean that all Cauchy sequences converges. So if we have an Cauchy…
Johan
  • 4,002
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Prove that $\|P_h\|\geq\int_{-1}^{1}|h(t)|dt$ using Tietze extension theorem

For each $h\in C([-1,1])$, define the linear functional $P_h:C([-1,1])\to \mathbb{R}$ by $$ P_h(f)=\int_{-1}^{1}f(t)h(t)dt,\qquad f\in C([-1,1]). $$ Here is the Banach space $C([-1,1]):=\{f:[-1,1]\to \mathbb{R}:f\textrm{ continuous}\}$, endowed with…
Hopeless
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Continuity of the inverse Operator

I was wondering whether the inverse Operator $A\mapsto A^{-1}$ is continuous on the set of continuously invertible linear operators $G\subset L(X)$, where X is supposed to be a Banach space (Can this may be weakend?). If this is not the case, do…
Aristo
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Version of Hahn-Banach Theorem

Problem: Let $X$ be a vector space over $\mathbb{R}$ and $Y \subset X$ a linear subspace. Let $p: X \to \mathbb{R}$ be a sublinear functional and $f: Y \to \mathbb{R}$ linear with $f \leq p$ on $Y$ (by Hahn-Banach this can be extended of…
Spaced
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identity operator isn't bounded

Suppose we consider the identity operator between the spaces $(C([0,1]),\| . \|_{\infty}) \rightarrow (C([0,1]),\| . \|_{1})$. Then the identity operator is bounded but its inverse isn't bounded. I am a little bit confused about this. So suppose we…
user329017
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Hilbert Space - Norm of derivative

If $H$ is a Hilbert space of entire functions with weighted norm $||f||^{2}=\int_{R} |\frac{f(t)}{g(t)}|^{2}dt$ for some entire function $g$ (not necessary in $H$). Can we find any relation between the norm of $f$ and the norm of it's derivative?…
user7630
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How to find the Hilbert Adjoint operator of $T(f)(t)=\int_{0}^tf(s)\ ds$?

Possible Duplicate: Finding the adjoint of an operator Consider the vector space $C[0, 1]$ with inner product, \begin{align*} \langle f, g\rangle=\int_{0}^1f(t)g(t)\ dt. \end{align*} Let $T:C[0, 1]\rightarrow C[0, 1]$ the bounded linear operador…
PtF
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Right Shift Operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T:H\rightarrow H$ such that $Te_n=e_{n+1}$, for $n=1, 2, \ldots.$ Find the range, null space, norm and…
PtF
  • 9,655
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Is the derivate on a closed subspace of $C^1[0,1]$ is a continuous linear map?

I'm trying to show that $D:(X, \|\cdot\|_\infty) \rightarrow C[0,1]$ is a continuous map. $D$ is the differential operator and $X$ is a closed (proper) subset of $C^1[0,1]$. The fact that $X$ is closed in $C^1[0,1]$ must be important in the proof…
Joe G.
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Matrix Representation of an Operator (Volterra)

Generally, how can I find the matrix of a given operator $K$ with respect to a given basis ${e_n}$? I thought I should use $\langle K(e_n),e_m\rangle$, is it the right way? In particular, how can I find the matrix representation of Volterra…
user
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