  It is well known that in the calculus of variations and in optimization there
exist many formulations of the fundamental propositions on the attainment of
the infima of sequentially weakly lower semicontinuous coercive functions on
reflexive Banach spaces. By either some constructive skills or the
regularization skill by inf--convolutions we show in this paper that all these
formulations together with their important variants are equivalent to each
other and equivalent to the reflexivity of the underlying space. Motivated by
this research, we also give a characterization for a normed space to be finite
dimensional: a normed space is finite dimensional iff every continuous
real--valued function defined on each bounded closed subset of this space can
obtain its minimum, namely the converse of the classical Weierstrass theorem
also holds true.
