Homework Assignment 6, MATH 515, Spring 09
Problem 14) (8 pts) Show that on every infinite dimensional normed space E
there exists a discontinuous (unbounded) linear functional. Hint: Use a Hamel
basis B⊂E,B={vi:i∈I},Isome index set, such that each v∈Ehas a
unique representation v=Pn
j=1 cjvijwith cjscalars.
Problem 15) (8 pts) Let Ebe a normed vector space and let λ∈E0such
that |λ|= 1. For any ε > 0 show that there is an xε∈Ewith |xε|= 1 and
λ(xε)>1−ε. Give an example to show that there need not be an x0∈Esuch
that |x0|= 1 and λ(x0) = 1. Hint: Consider `1and `∞from Problem 12.
Problem 16) (8 pts) page 91, Chapter IV, §6, Exercise 6