Exercise 4, § 1

a) Let $B_n$ be the closed unit ball in the Euclidean space $\mathbf{R}^n$, and let $A_n$ be the set of functions $f \in \mathscr{C}(B_n; \mathbf{R})$ of the form

with $|\lambda| \leq 1/n$ and $\|a\| \leq 1/n$. The set $A_n$ is convex and compact in the Banach space $\mathscr{C}(B_n; \mathbf{R})$, and possesses the following property: for any functions $f_1, \ldots, f_n$ belonging to $A_n$, there exists a point $x \in B_n$ such that $f_1(x) = \ldots = f_n(x) = 0$; however, there exist $n+1$ functions $g_1, \ldots, g_{n+1}$ belonging to $A_n$ that are not simultaneously zero at any point of $B_n$ (one may consider the mapping $x \mapsto x/(1-\|x\|^2)$ of $\overset{\circ}{B}_n$ into $\mathbf{R}^n$). b) Let $B'_n$ be the set $B_n$ equipped with the discrete topology, and let $D_n = \beta B'_n$. Let Y (resp. $Y'$) be the locally compact topological sum space of the spaces $D_n$ (resp. $B'_n$). Show that $\beta Y = \beta Y'$. c) For every integer $n$, let $K_n \subset \mathscr{C}(D_n; \mathbf{R})$ be the set of all real-valued functions whose restriction to $B'_n = B_n$ belongs to $A_n$. Let K be the subset of $\mathscr{C}(\beta Y; \mathbf{R})$ formed by the functions whose restriction to $D_n$ belongs to $K_n$ for every $n \in \mathbf{N}$. Show that K is a compact convex subset of the Banach space $\mathscr{C}(\beta Y; \mathbf{R})$. For every finite family $(f_i)_{1 \leq i \leq n}$ of functions in K, let $W(f_1, \ldots, f_n)$ be the set of $y \in Y'$ such that $f_i(y) = 0$ for $1 \leq i \leq n$. Show that the $W(f_1, \ldots, f_n)$ form a base of a filter $\mathfrak{W}$ on the discrete space $Y'$. From this, conclude that there exists a point $\omega \in \beta Y - Y$ such that the trace on $Y'$ of the filter of neighborhoods of $\omega$ in $\beta Y$ is a filter finer than $\mathfrak{W}$. Setting $X = \beta Y - \{ \omega \}$, deduce from this that $K \subset \mathscr{K}(X; \mathbf{R})$ and, with the help of Exercise 2, conclude that in $\mathscr{K}(X; \mathbf{R})$ the set K is convex and compact, but not strictly compact.