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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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Decomposing a function into $L^p+L^q$

I'm trying to prove that $\vert x \vert^{-1}$ can be decomposed into an $L^p+L^q$ functions, where $p<3$ and $q>3$, the integration is over $\mathbb{R}^3$, but I'm still unable to proof/ understand this. The I'm going on about it is, fix $r>0$, so…
SamKC71
  • 17
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Functional Analysis- Brezis Excercise 1.16 .

Let $ E = l^{1} $, so that $ E^{*} = l^{\infty} $ Consider: $ N = c_0 = \{x= (x_k): \lim\limits_{k\to \infty}x_k = 0\} $ as a closed subspace of $ l^{\infty}$ . Determine: $ N^\bot=\{x\in E : \langle f, x\rangle = 0 \quad \forall f\in N \}…
Gabriel Granda
  • 129
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A sufficient and necessary condition for the boundedness of linear operator

Let $X$ and $Y$ be two normed linear spaces and $T: X \rightarrow Y$ is a linear operator(mapping). Prove that $T$ is bounded if and only if it maps weakly convergent sequences to weakly convergent sequences. To my best knowledge, I think it's easy…
swj
  • 35
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Perturbation of Laplace operator

Let $V_1$ and $V_2\in L^2(\mathbb{R}^3)$ be real-valued functions, then $V_1(x_1)$ and $V_2(x_2)$ can be viewed as multiplicative operators in $L^2(\mathbb{R}^6)$ by $V_i(u) = V_i(x_i)u(x_1,x_2)$. Prove that $-\Delta+V_1(x_1)+V_2(x_2)$ is…
Will Morry
  • 31
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Theorem 3.10 in Rudin's Functional Analysis

Theorem 3.10: Suppose $X$ is a vector space and $X'$ is a separating vector space of linear functionals on $X$. Then the $X'$-topology $\tau'$ makes $X$ into a locally convex space whose dual space is $X'$. In addition to having the same trouble…
user193319
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Example of not continuous map with closed graph

Suppose that X={f $\in$ C([$0,1$]): $\exists \delta>0$ f|[$0,\delta$]=$0$}, $||f||=sup${$|f(t)|: t \in [0,1]$} and T: X -> X, (Tf)(x)=$\frac{1}{x}$f(x), for $x \in [0,1]$. Prove that T has closed graph but is not continuous.
Ana304
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traces on matrices with enteries in quaternions

I am wondering if there exists any quaternionic valued trace $T:M_{n}(\mathbb{H}) \rightarrow \mathbb{H}$ on a quaternionic matrix? Where the trace is a linear functional on a C*-algebra with the property T(AB)=T(BA). generally, quaternions are not…
Emilly
  • 65
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Find a optimal Sobolev Space $H^s$

It is a homework problem. Function $f:\mathbb{R}^2\rightarrow\mathbb{R}$,$\Omega$ is a disk at origin with radius as $\dfrac{1}{2}$. $f(x) = \log(\log(\dfrac{1}{|x|}))$, where $x\in\Omega$(i,e. $|x|\in (0,1/2)$), it is easy to show that $f\in…
Yimin
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Uniqueness of the equation $Ax = y$ in a Hilbert spaece

Suppose $H$ is a Hilbert space and $A \in B(H)$ satisfies $\langle Ax, y\rangle = \langle x, Ay\rangle , \, \forall x, y \in H$. (Notice that this means $A = A^*$) Prove: If $\DeclareMathOperator*{\range}{range} \range(A)$ is dense in $H$, then for…
U2647
  • 583
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Bidual space of a normed space

I have trouble understanding the bidual space of the normed space $(E, \|\cdot \|)$. Could someone help me understand it?
Giovanni
  • 184
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Graph norm of a closed operator

Let's say we have a separable Banach space ($B$, $\Vert \cdot \Vert$). $A: D(A) \to B$ is a closed operator defined on a linear subspace $D(A)$ of $B$. Define the graph norm on $D(A)$ by $$\Vert x \Vert_{D(A)}= \Vert x \Vert + \Vert Ax \Vert.$$ It…
lye012
  • 383
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4 answers

Solve $f(x)=c \times f(\frac{x}{2})$ for $c$

Given: Function $f(x)$ is infinitely differentiable equation (1) $f(x)=c \times f(\frac{x}{2})$ We have to find all $c$, for which the (1) has non-zero solutions Any hints on theorems to apply here, I reckon it's somehow related to ODEs
user119510
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These two norms of $C^k[a,b]$ are equivalent.

In $C^k[a,b]$ we can define $$ \|f\|_* := \|f\|_\infty + \left\|f^{(k)}\right\|_\infty $$ and $$ \|f\|_{**} := \sum_{j=0}^k \left\|f^{(j)}\right\|_\infty$$ It is obvious that $\|f\|_*\leq \|f\|_{**}$. However, how can I show that $K\|f\|_*\geq…
Rub
  • 385
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1 answer

linear transformation between spaces of continuous function on metric space

Let $M_1$ and $M_2$ be compact metric spaces. Denote $C(M_i)$ as the space of continuous functions from $M_i$ to $\Bbb C$ with supremum-norm, $i=1,2$. A linear function $T:C(M_1)\to C(M_2)$ is said to be positive if $T(f) \ge 0$ whenever $f \ge 0$…
Lawrence
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