Math 511 Notes
Mathematical Analysis II
Dr. John Travis
Mississippi College
CHAPTER THREE
Defn: A metric space will
describe a set X with a function r
so that for any two points (x,y), there corresponds a real number r(x,y)
such that:
-
r(x,y) > 0 and r(x,y)=0
if and only if x=y
-
r(x,y) = r(y,x)
-
r(x,z) < r(x,y)
+ r(y,z), for any
z
Examples of metric spaces: (see page
92+)
-
n-dimensional reals with the Euclidean metric
-
sequence space l
oo
-
sequence space l
1
-
sequence space c
-
C([a,b]) = continuous functions defined on
the interval [a,b] with r(f,g)
= max | f(x)-g(x) |
Lp Spaces:
Given a positive number p, denote by Lp(X) the collection of
functions defined on X such that |f|p is integrable. Define
the "p-norm" of a given function f to be:
|| f ||p = {,/` |f|p
dm }1/p
Similarly, define L00(X) to
be the collection of measurable and essentially bounded functions. Define
the "00-norm" of a given function to be:
|| f ||p = essential supremum
|f|
Holder's Inequality: Assume 1/p
+ 1/q = 1. If f e Lp(X)
and g e Lq(X),
then f g e L1(X)
and || f ||1 < || f ||p || g ||q
Pf: See page 96
Minkowski's Inequality: Let 1 <
p < 00. Then, f,g e Lp(X)
implies f+g e Lp(X)
and || f+g ||p < || f ||p + || g ||p.
Pf: See page 97
Theorem 3.2.3: If 1 <
p < 00, then Lp(X) is a complete metric space.
Pf: See page 98