Page 340 - Advanced Linear Algebra

324 Advanced Linear Algebra

% ( ( & ( (
"~ Á # ~
? ° @ °
¨
and then sum on to deduce Holder's inequality.

38. Prove Minkowski's inequality
° ° °
B B B
%b & ( (  % ( ( b & (
8 ( 9 8 9 8 9
~ ~ ~
as follows:
a Prove it for ~ first.
)
)
b Assume € . Show that
( (%b & ( (  % ( (%b & c ( ( b & ( (%b & c
c Sum this from ~ to and apply Holder's inequality to each sum on
¨
)

the right, to get

( %b & (
~
° ° °

 H8 % ( 9 b 8 ( ( 9 I8 ( ( % & b & ( 9
~ ~ ~
Divide both sides of this by the last factor on the right and let ¦B to
deduce Minkowski's inequality.
39. Prove that is a metric space.

M