Sequence spaces of continuous functions (Q5948873)

A Banach space lying in \(C(X)\), where \(X\) is a compact \(T_2\)-space, along with the following properties is termed as Banach function space:NEWLINENEWLINENEWLINEi) if \(x\), \(x_2\) are distinct points of \(X\), then \(f(x_1)\neq f(x_2)\) for some function \(f\) in the function space;NEWLINENEWLINENEWLINEii) there is a positive constant \(C\) such that the function space norm on \(f\) is at least \(C\|f\|_\infty\) where \(\|f\|_\infty=\sup\{|f(x)|:x\in X\}\).NEWLINENEWLINENEWLINEFor a Banach function space \(E\), \(\widetilde E\) shall denote the Banach space of all bounded sequences \({\widetilde C}=\langle f_k\rangle^\infty_{k=1}\) of functions \(f_1,f_2,\dots\) in \(E\) under component-wise operations and the norm \(\|\widetilde f\|=\text{Sup}\{\|f_k\|:k\in N\}\); it is seen that \(\widetilde E\) can be considered as a Banach space lyig in \(C(\widetilde x)\).NEWLINENEWLINENEWLINEThe paper gives developments regarding the systematic study of \(\widetilde E\) and also depicts the line of developments regarding the main application of \(\widetilde E\).