Hamel Bases, Convexity and Analytic Sets in Fréchet Spaces

It is shown that a Hamel basis over the field of reals of an infinite dimensional linear Polish space can not be an analytic set. Furthermore, if

is an infinite linearly independent subset of a Fréchet space

X X

and if

C C

is the convex cone generated by

(x_{\alpha}), ( x α ) ,

then

C C

is not a closed set. In particular, the convex cone generated by a Hamel basis in such a space can not be closed.The notion of convex and midpoint convex functions extended to the case when the domain of the functions is a connected open set, and analytic graph theorems are given for these functions. It is shown also that if

f:{\mathbb R}^n \to {\mathbb R} f : R n → R

is an order monotone function, then

f f

is Baire measurable, but in general,

f f

is not universally measurable