Page 319 - Advanced Linear Algebra
P. 319
Metric Spaces 303
)
%
2 For , let be the set of all sequences ~ ²% ³ of real or complex)
M
²
numbers for which
B
( % ( B
~
We define the -norm of by
%
°
B
)) ~ % 8 % ( ( 9
~
Then is a metric space, under the metric
M
°
B
)
² Á ³ ~ % c & ) ~ ( % c & (
%&
8 9
~
The fact that is a metric follows from some rather famous results about
M
sequences of real or complex numbers, whose proofs we leave as well-
(
)
hinted exercises.
Holder's inequality Let Á and b ~ . If % M and & M ,
¨
then the product sequence %& ~²% & ³ is in and
M
) )%& ) ) % ) ) &
that is,
° °
B B B
%& ( 8 ( ( % 9 8 ( ( & ( 9
~ ~ ~
(
)
A special case of this with ~ ~ 2 is the Cauchy–Schwarz inequality
B B B
%& ( m ( ( ( % m ( & (
~ ~ ~
Minkowski's inequality For , if %& M then the sum % b &
Á
~²% b & ³ is in and
M
) )% ) & b ) ) % & ) b
that is,
° ° °
B B B
%b & ( ( % ( ( b & (
8 ( 9 8 9 8 9
~ ~ ~
