 
Summary: Lecture 29: Theorem 1.11 and Minkowski's inequality
1. Proof of Theorem 1.11
This note contains the proof of the nontrivial part of theorem 1.11. In order to
keep things a bit simpler we only consider the case where V is a real vector space.
Surprisingly, the proof is not completely within the domain of Linear Algebra. It
requires some facts which you may have seen in Calculus but which are usually
proved in a Real Analysis lecture (such as MTH 320). The proof is taken from K.
Yosida's book on Functional Analysis, Chapter I.5.
Theorem: Assume that V is a normed real vector space such that the parallelo
gram identity holds for all vectors x, y, i.e.
x + y 2
+ x  y 2
= 2( x 2
+ y 2
).
Then there is an inner product ( . , . ) on V such that (x, x) = x 2
.
Proof:
We define1
(1) (x, y) =
