Sequential Closure of the Space of Jointly Continuous Functions in the Space of Separately Continuous Functions

Given compact spaces X and Y, we study the space S(X × Y) of separately continuous functions f : X × Y → ℝ endowed with the locally convex topology generated by the seminorms fx=maxy∈Yfxy,x∈X,andfy=maxx∈Xfxy,y∈Y.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{array}{lll}{\left\Vert f\right\Vert}^x={ \max}_{y\in Y}\left|f\left(x,y\right)\right|,\kern0.5em x\in X,\hfill & \mathrm{and}\hfill & {\left\Vert f\right\Vert}_y={ \max}_{x\in X}\left|f\left(x,y\right)\right|,\kern0.5em y\in Y.\hfill \end{array} $$\end{document}