Articles of functional analysis

Distance from a weak* closed subspace is achieved by an element?

Let $X$ denotes a separable Banach space and $X^{*}$ its dual space. Let $M\subseteq X^{*}$ be a weak* closed subspace of $X^{*}.$ Let $\varphi\in X^{*}\setminus M$ be given. I have the following question: is it true that there exists a $\psi\in M$ such that $$\inf\{||\varphi-m||:m\in M\}=||\varphi-\psi||,$$ where $||.||$ denotes the operator norm of linear functionals […]

Compatibility of topologies and metrics on the Hilbert cube

Consider the Hilbert cube $Y = [0,1]^\mathbb{N}$. It is easy to define four classes of metrics on $Y$ for $\gamma>0$ and $\omega>1$: $$d^\gamma_{sup,pol}(x,y) = \sup_{k\geq 1} |x_k-y_k|/k^\gamma,$$ $$d^{\gamma}_{sum,pol}(x,y) = \sum_{k=1}^\infty |x_k-y_k|/k^{1+\gamma}, $$ $$d^\omega_{sup,exp}(x,y) = \sup_{k\geq 1} |x_k-y_k|/\omega^k,$$ $$d^\omega_{sum,exp}(x,y) = \sum_{k=1}^\infty |x_k-y_k|/\omega^k $$ Question, do all these metrics define the same topology? does any of this […]

Contradiction achieved with the Pettis Measurability Theorem?

$\bf{\text{(Pettis Measurability Theorem)}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure. The following are equivalent for $f:\Omega\to X$. (i) $f$ is $\mu$-measurable. (ii) $f$ is weakly $\mu$-measurable and $\mu$-essentially separately valued. (iii) $f$ is Borel measurable and $\mu$-essentially separately valued. $\bf{\text{Relevant Definitions:}}$ A function $f:\Omega\to X$ is simple if it assumes only finitely many values. That is, […]

Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition – that is, either that $V$ maps bounded sets to precompact sets, or equivalently, that for any bounded sequence $(f_n)$ in the domain, $\{Vf_n\}$ has a convergent subsequence. I have seen an […]

Norm of the Resolvent

Let $\mathbb{H}$ be a Hilbert space, $A$ a self-adjoint operator with domain $D_{A}$, $R_{A}$ the resolvent of $A$, and $z$ a point in the resolvent set $\rho(A)$. How could you prove the inequality \begin{equation} ||R_{A}(z)|| \leq 1/ d(z,\sigma(A)), \end{equation} where $\sigma(A)$ is the spectrum of $A$, and $d(z,\sigma(A))$ the distance of $z$ from $\sigma(A)$? I […]

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different proof that only works when $X$ is sigma finite, but then it establishes also that the dual of $L^1(X)$ is […]

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions of four variables. Then, “because the range function depends on 4 […]

About Cauchy sequence

Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{2n-1})$ and $(a_{2n})$ has no Cauchy subsequence. Is it also true that $(a_n)$ has no Cauchy subsequence? Let $A=\{a_{2n}:n\in\mathbb{N}\}$, is it true every Cauchy sequence in $A$ is constant?

If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?

$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse. I think the answer is yes, the only thing I am unable to show is what is the inverse of $T^*$ and if it is continuous?

Is the composition function again in $L^2$

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator $T:L^2[a,b] \rightarrow L^2[a,b]$ which is defined by $Tf = f \circ g$, is a bounded linear operator?