A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space.

A Banach space B is isometric with a subspace of its second conjugate space B** under the "natural mapping" for which the element of B** which corresponds to the element xo of B is the linear functional Fxz defined by Fxs(f) = f(xo) for each f of B*. If every F of B** is of this form, then B is said to be reflexive and B is isometric with B** under this natural mapping. The purpose of this note is to show that B can be isometric with B** without being reflexive. The example given to show this is a space isomorphic with a Banach space known to not be -reflexive, but to be isomorphic with its second conjugate space.1 A sequence {x"} of elements of a Banach space B is said to be a basis for B if for each x of B there is a unique sequence of numbers Ia". such m n that x a x in the sense that lim llx Eatx'II = 0. A fundamental 1 n 1 sequence {x"} is a basis if and only if there is a positive number e such that n+p n 1 aJ E axjl 2 ell E aixfll for any positive integers n and p and numbers 1 1 ta}i.2 If e = 1, the basis will be called an orthogonal basis. But for any n eD basis {x"}, Illxlll = sup,llEatx'II for x = EatxI defines a new norm 111