Advanced Analysis: Outline
Math 508, Fall 2010
Jerry L. Kazdan
This outline of the course is to help you step back and get a larger view of what we have
done. Since this is only an outline, I will often not explicitly state the precise assumptions
needed for the assertion to hold.
Important Sets
R
: T
HE
R
EAL
N
UMBERS
Primary features: they form a
field
, are
ordered
,
and are
complete
. The completeness is the primary feature that distin
guishes from the much smaller field
Q
of rational numbers. The function

x

is a
norm
that measures the size of a real number.
Basic concepts:
convergence
of a sequence,
limit point
of a sequence
and of a set,
open
,
closed
, and
bounded
sets,
Cauchy sequences
,
count
able
and
uncountable
sets.
Two important theores are the Bolzano
Weierstrass Theorem (bounded sequences have convergent subsequences)
and the HeineBorel Theorem (when open covers have a finite subcover).
These theorems are associated with the
compactness
of a set.
C
: C
OMPLEX NUMBERS
:
z
=
x
+
iy
They also form a complete field but
are not ordered. The norm

z

:
=
x
2
+
y
2
measures the size. The trian
gle inequality

z
+
w
 ≤ 
z

+

w

is basic. The concepts of convergence
and limit point etc. extend immediately. The convergence of an
infinite
series
arises frequently. We define
e
z
, cos
z
, and sin
z
using power series.
Euler’s beautiful observation
e
ix
=
cos
x
+
i
sin
x
is valuable.
G
ENERALIZATION
:
R
k
,
C
k
,
1
,
2
,
the set of
n
×
k
real or complex
matrices. These are important
normed linear spaces
. The norm
X
is
used to define a
metric d
(
X
,
Y
)
:
=
X

Y
.
The concepts of convergence and limit points generalize immediately, as
do those of open and closed sets, Cauchy sequence, compactness, etc. If
A
is a square matrix we define
e
A
using the power series.
1
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The concept of
connectedness
arises and is important.
These spaces are all complete (proving the completeness of
1
and
2
takes some work). Subsets of these, such as the closed unit ball of points
X
where
X
≤
1 are also metric spaces, although they are not linear
spaces (if
X
and
Y
are in the unit ball,
X
+
Y
might not be). Most of our
metric spaces will simply be subsets of normed linear spaces.
The point of introducing the examples
1
and
2
is that while closed
bounded sets are compact in
R
k
and
C
k
, they are usually
not
compact in
1
or
2
– or in other important examples (such as the set of continuous
functions on
[
0
,
1
]
with the uniform norm) that we will meet soon.
In some normed linear spaces there is an
inner product
X
,
Y
and the
norm is given by
X
:
=
X
,
X
. These spaces are particularly easy
to use – and arise frequently in applications.
In them one can define
two vectors
X
and
Y
to be
orthogonal
if
X
,
Y
=
0, in which case the
Pythagorean theorem holds:
X
+
Y
2
=
X
2
+
Y
2
.
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 Fall '10
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 Math, Topology, Derivative, Continuous function, Metric space, Compact space

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