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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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Properties of an integral operator: $(Au)(x)=\int_{\bf R}e^{-|x|-|y|}u(y)dy$

My questions are motivated by the following exercise: Consider the eigenvalue problem $$ \int_{-\infty}^{+\infty}e^{-|x|-|y|}u(y)dy=\lambda u(x), x\in{\Bbb R}.\tag{*} $$ Show that the spectrum consists purely of eigenvalues. Let $A:L^2({\Bbb…
user9464
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Integration in Hilbert space

Let $H$ and $K$ be two Hilbert spaces equipped with orthonormal bases $\{h_{i,j}\}_{i,j\in\mathbb{Z}}$ and $\{k_{i,j}\}_{i=0,1,2,\ldots,\ j\in\mathbb{Z}}$ respectively and $\mu$ a positive real number smaller than $1$. Suppose $n$ and $\gamma$ are…
Dastan
  • 328
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Intuition of Mercer's theorem?

I have recently been trying to gain some intuition about Mercer's Theorem: $$K(x,y)=\sum_{i=1}^{\infty}\lambda_ie_i(x)e_i(y)$$ According to this video, each symmetric, positive semi-definite kernel constitutes an inner product in some associated…
J.Galt
  • 961
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1 answer

Differential Gateaux

I want to take a Gateaux differential of a functional $$I(u)=\int_{\Omega} \left[\frac{1}{2}(\frac{du}{dx})^2+\frac{1}{2}u^2\right]d{\Omega}$$ so the Gateaux differential is defined as follow: $$D_hI(u)=[\frac{d}{dw}I(u+hw)]_{w=0}$$ My…
John G.
  • 125
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2 answers

closed linear subspace of incomplete inner product space

Let $M$ be a closed linear subspace of an incomplete inner product space $X$ and let $M + M^\perp \neq X$ then is it true that $M \neq M^{\perp\perp}$. If true then how to prove it and if not then do we have a counterexample. I know the converse of…
m pandey
  • 309
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Let $X$ be a normed space. If $(x_n)$ is a Cauchy sequence with $x_n \overset{w}{\rightarrow} 0$, then $x_n \to 0$

The following is the problem 3.35 of the book Banach Space Theory from Fabian, Habala, et al. Let $X$ be a normed space. If $(x_n)$ is a Cauchy sequence with $x_n \overset{w}{\rightarrow} 0$, then $x_n \to 0$. $x_n \overset{w}{\rightarrow} 0$ means…
user 242964
  • 1,898
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2 answers

Is there some reason that the fact that $L^\infty$ is the dual space of $L^1$ is an important fact?

Why is the fact that $L^\infty$ is the dual space of $L^1$ an important fact?
David
  • 635
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Question about a paper on approximate identities

I am currently reading this paper on approximate identities of ternary Banach algebras. Assume that $(A, [.,.,.])$ is a ternary Banach algebra. A bounded net $(e_{\alpha}, f_{\alpha})$ is said to be left-bounded approximate identity for $A$ if…
Math Lover
  • 3,602
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for a 'non-reflexive' space, the weak star topology is strictly coarser than the weak topology on the dual space

I'm trying t0 work out the fine details of a claim in "Functional analysis, Sobolev spaces and PDE's" by H. Brezis. Let $J: \begin{cases} E\longrightarrow E^{**} & \\ x \longmapsto J(x) & \end{cases}$, with $J(x)\left ( f \right )= f(x),…
user58664
  • 593
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1 answer

basis of $\{f\in C^2([a,b]):f(a)=f(b)=0 \}$

Regarding the Dirichlet Laplacian operator $L^{(D)}f=-\frac{d^2}{dx^2}f$ on the set $D(L^{(D)})=\{f\in C^2([a,b]):f(a)=f(b)=0 \}$ on the Hilbert space $L^2(a,b)$ one can show that it is essentially self-adjoint by showing that the family of vectors…
mathemagician99
  • 904
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Spectrum theorem the link between cyclic vector and approximate eigenvector

The multiplication version of spectrum theorem says that for a bounded self-adjoint operator $A$ on $H$, there exists measures $\{\mu_n\}$ and a unitary operator $U : H\rightarrow \oplus_{n=1}^{n} L^2(\mathbb{R},\mu_n)$ s.t.…
89085731
  • 7,614
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$T:C[0,1]\to C[0,1]$ defined by $Tx(t)=\int_0^tx(s)ds,t\in[0,1]$ . How to show $T^{-1}$ is unbounded.

$T:C[0,1]\to C[0,1]$ defined by $Tx(t)=\displaystyle\int_0^tx(s)ds,t\in[0,1]$ . How to show $T^{-1}$ is unbounded. First off all in the book the norm is not mentioned so I suppose (I think) the norm is supnorm. My intuition says that find a function…
Jale'de jaled
  • 780
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1 answer

Proof checking : If $\{h\in\mathcal{H}:||h||\leq1\}$ is compact, then $\text{dim}\mathcal{H}<\infty$.

I am going through the functional analysis book by John B. Conway. I have a problem If $\{h\in\mathcal{H}:||h||\leq1\}$ is compact, show that $\text{dim}\mathcal{H}<\infty$. Where $\mathcal{H}$ is Hilbert space. My proof is following : Let…
Jingeon An-Lacroix
  • 1,635
2
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1 answer

weakly convergent sequence converges under a compact operator

Let $H$ be a Hilbert space and $T\in\mathcal{K}(H)$. Show that if $(x_n)_{n\in\mathbb{N}}$ is a sequence in $H$ that converges weakly to $x_0\in H$ then $\lim_{n\to\infty}||Tx_n-Tx_0||=0$. My proof: Since $\overline{T(B_1)}$ is compact and thus…
mathemagician99
  • 904
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1 answer

Prove that $\forall \delta>0,\delta B_Y\subseteq T(B_X)\Leftrightarrow \gamma \delta B_Y \subseteq\gamma T(B_X) \quad \forall \gamma >0 $

Let $X,Y$ normed vectorial space and $T\in B(X,Y).$ Prove that $$\forall \delta>0,\delta B_Y\subseteq T(B_X)\Leftrightarrow \gamma \delta B_Y \subseteq\gamma T(B_X) \quad \forall \gamma >0 $$ My attempt: If $\quad \forall \delta >0 \quad\delta B_Y…
Giulia B.
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