ANALYSIS TOOLS WITH APPLICATIONS
1
1.
Introduction
Not written as of yet. Topics to mention.
(1) A better and more general integral.
(a) Convergence Theorems
(b) Integration over diverse collection of sets. (See probability theory.)
(c) Integration relative to di
ff
erent weights or densities including singular
weights.
(d) Characterization of dual spaces.
(e) Completeness.
(2) In
fi
nite dimensional Linear algebra.
(3) ODE and PDE.
(4) Harmonic and Fourier Analysis.
(5) Probability Theory
2.
Limits, sums, and other basics
2.1.
Set Operations.
Suppose that
X
is a set. Let
P
(
X
)
or
2
X
denote the power
set of
X,
that is elements of
P
(
X
) = 2
X
are subsets of
A.
For
A
∈
2
X
let
A
c
=
X
\
A
=
{
x
∈
X
:
x /
∈
A
}
and more generally if
A, B
⊂
X
let
B
\
A
=
{
x
∈
B
:
x /
∈
A
}
.
We also de
fi
ne the symmetric di
ff
erence of
A
and
B
by
A
4
B
= (
B
\
A
)
∪
(
A
\
B
)
.
As usual if
{
A
α
}
α
∈
I
is an indexed collection of subsets of
X
we de
fi
ne the union
and the intersection of this collection by
∪
α
∈
I
A
α
:=
{
x
∈
X
:
∃
α
∈
I
3
x
∈
A
α
}
and
∩
α
∈
I
A
α
:=
{
x
∈
X
:
x
∈
A
α
∀
α
∈
I
}
.
Notation 2.1.
We will also write
`
α
∈
I
A
α
for
∪
α
∈
I
A
α
in the case that
{
A
α
}
α
∈
I
are pairwise disjoint, i.e.
A
α
∩
A
β
=
∅
if
α
6
=
β.
Notice that
∪
is closely related to
∃
and
∩
is closely related to
∀
.
For example
let
{
A
n
}
∞
n
=1
be a sequence of subsets from
X
and de
fi
ne
{
A
n
i.o.
}
:=
{
x
∈
X
: #
{
n
:
x
∈
A
n
}
=
∞
}
and
{
A
n
a.a.
}
:=
{
x
∈
X
:
x
∈
A
n
for all
n
su
ﬃ
ciently large
}
.
(One should read
{
A
n
i.o.
}
as
A
n
in
fi
nitely often and
{
A
n
a.a.
}
as
A
n
almost al
ways.) Then
x
∈
{
A
n
i.o.
}
i
ff
∀
N
∈
N
∃
n
≥
N
3
x
∈
A
n
which may be written
as
{
A
n
i.o.
}
=
∩
∞
N
=1
∪
n
≥
N
A
n
.
Similarly,
x
∈
{
A
n
a.a.
}
i
ff
∃
N
∈
N
3
∀
n
≥
N, x
∈
A
n
which may be written as
{
A
n
a.a.
}
=
∪
∞
N
=1
∩
n
≥
N
A
n
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
BRUCE K. DRIVER
†
2.2.
Limits, Limsups, and Liminfs.
Notation 2.2.
The Extended real numbers is the set
¯
R
:=
R
∪
{±
∞
}
,
i.e.
it
is
R
with two new points called
∞
and
−∞
.
We use the following conventions,
±
∞
·
0 = 0
,
±
∞
+
a
=
±
∞
for any
a
∈
R
,
∞
+
∞
=
∞
and
−∞ − ∞
=
−∞
while
∞ − ∞
is not de
fi
ned.
If
Λ
⊂
¯
R
we will let
sup
Λ
and
inf
Λ
denote the least upper bound and greatest
lower bound of
Λ
respectively. We will also use the following convention, if
Λ
=
∅
,
then
sup
∅
=
−∞
and
inf
∅
= +
∞
.
Notation 2.3.
Suppose that
{
x
n
}
∞
n
=1
⊂
¯
R
is a sequence of numbers. Then
lim inf
n
→∞
x
n
= lim
n
→∞
inf
{
x
k
:
k
≥
n
}
and
(2.1)
lim sup
n
→∞
x
n
= lim
n
→∞
sup
{
x
k
:
k
≥
n
}
.
(2.2)
We will also write
lim
for
lim inf
and
lim
for
lim sup
.
Remark
2.4
.
Notice that if
a
k
:= inf
{
x
k
:
k
≥
n
}
and
b
k
:= sup
{
x
k
:
k
≥
n
}
,
then
{
a
k
}
is an increasing sequence while
{
b
k
}
is a decreasing sequence. Therefore the
limits in Eq. (2.1) and Eq. (2.2) always exist and
lim inf
n
→∞
x
n
= sup
n
inf
{
x
k
:
k
≥
n
}
and
lim sup
n
→∞
x
n
= inf
n
sup
{
x
k
:
k
≥
n
}
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Three '10
 Smith
 Topology, Sets, Probability, lim, Metric space, Open set, Topological space, BRUCE K. DRIVER

Click to edit the document details