Uniform approximation and a generalized minimax theorem (Q1066388)

Let X be a compact Hausdorff space and Y a normed linear space with norm \(\| \|\). Let C(X,Y) be the set of all continuous functions from X to Y. This paper characterizes best approximations \(f^*\) of F by elements f of a finite-dimensional subspace A of C(X,Y) in the uniform topology. In other words, the characterization of those elements \(f^*\) of A which minimize \(\max_{x\in X}\| f(x)-F(x)\|\) over A, is treated. The results extend the well-known classical ones which are valid in C(X). The uniqueness of best approximation is also presented by means of a Haar condition. In order to derive these, a generalized minimax theorem is formulated and proved. It is shown that one of its corollaries is useful to the above approximation problem that can be considered as a minimax optimization problem. It takes on the following form: \[ \min_{u\in U}\sup_{v\in V}J(u,v)=\sup_{({\bar \lambda},\bar v)\in \bar V}\min \quad_{u\in U}\sum^{n+1}_{i=1}\lambda_ iJ(u,v_ i), \] where V is an arbitrary set, U is an n-dimensional compact convex set, J(\(\cdot,v): U\to R\) is an l.s.c. and convex function for each \(v\in V\), and the set \(\bar V\) is defined by \[ \bar V=\{({\bar \lambda},\quad \bar v)| {\bar \lambda}=(\lambda_ 1,...,\lambda_{n+1}),\bar v=(v_ 1,...,v_{n+1}),\sum^{n+1}\quad_{i=1}\lambda_ i=1,\quad \lambda_ i\geq 0,\quad v_ i\in V(i=1,...,n+1)\}. \]