Goldstine theorem
In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows: The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, c0 spaceand its bi-dual space Lp spaceδ > 0 ,x ∈ ( 1 + δ )so it suffices to show that the intersection is nonempty.Assume for contradiction that it is empty.and by the Hahn–Banach theorem there exists a linear form= 0 , φ ( x ) ≥ 1 + δφ ∈ span 1 + δ ≤ φ ( x ) =Examine the setbe the embedding defined byEv{\displaystyle J(x)={\text{Ev}}_{x},}Ev{\displaystyle {\text{Ev}}_{x}(\varphi )=\varphi (x)}Sets of the formform a base for the weak* topology,[2] so density follows once it is shownEv{\displaystyle {\text{Ev}}_{x}\in U.}Ev{\displaystyle {\text{Ev}}_{x}\in (1+\delta )J(B)\cap U.}Ev{\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B).}The goal is to show that for a sufficiently smallEv{\displaystyle {\frac {1}{1+\delta }}{\text{Ev}}_{x}\in J(B)\cap U.}Directly checking, one hasNote that one can choosesufficiently large so that