A
Metrics, Norms, Inner Products, and Topology
These appendices collect the background material needed for the main part of
the volume. In keeping with the philosophy of this text, we formulate these as
“minicourses” with the goal of providing substantial, though not exhaustive,
introductions to and reviews of their respective subjects. Topics that are not
typically part of standard beginning mathematics graduate courses are given
more detailed attention, while other results are either formulated as exercises
(often with hints) or stated without proof. Sources for additional information
on the material of this and most of the other appendices include Folland’s real
analysis text [Fol99], Conway’s functional analysis text [Con90], and the the
operator theory/Hilbert space text [GG01] by Gohberg and Goldberg.
A.1 Notational Conventions
We first review some of the notational conventions that are used throughout
this volume.
Unless otherwise specified, all vector spaces are taken over the complex
field
C
.
In particular, functions whose domain is
R
d
(or a subset of
R
d
) are
generally allowed to take values in the complex plane
C
.
Integrals with unspecified limits are taken over either the real line or
R
d
,
according to context. In particular, if
f
:
R
→
C
,
then we take
integraldisplay
f
(
x
)
dx
=
integraldisplay
∞
−∞
f
(
x
)
dx.
The extended real line is
R
∪{−∞
,
∞}
= [
−∞
,
∞
]
.
We use the conventions
that 1
/
0 =
∞
, 1
/
∞
= 0
,
and 0
·∞
= 0
.
If 1
≤
p
≤∞
is given, then its
dual index
or
dual exponent
is the extended
real number
p
′
that satisfies
1
p
+
1
p
′
= 1
.
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A Metrics, Norms, Inner Products, and Topology
Explicitly,
p
′
=
p
p
−
1
.
The dual index lies in the range 1
≤
p
′
≤∞
,
and we have 1
′
=
∞
, 2
′
= 2
,
and
∞
′
= 1
.
The
Kronecker delta
is
δ
ij
=
braceleftBigg
1
,
i
=
j,
0
,
i
negationslash
=
j.
A.2 Metrics and Convergence
A metric determines a notion of distance between points in a set.
Definition A.1 (Metric Space).
Let
X
be a set. A
metric
on
X
is a function
d:
X
×
X
→
R
such that for all
f, g, h
∈
X
we have:
(a) d(
f, g
)
≥
0
,
(b) d(
f, g
) = 0 if and only if
f
=
g,
(c) d(
f, g
) = d(
g, f
)
,
and
(d) the
Triangle Inequality
: d(
f, h
)
≤
d(
f, g
) + d(
g, h
)
.
In this case,
X
is a called a
metric space
. The value d(
f, g
) is the
distance
from
f
to
g.
If we need to explicitly identify the metric we write “let
X
be a metric
space with metric d” or “let (
X,
d) be a metric space.”
A metric space need not be a vector space, although this will be true of
most of the metric spaces encountered in this volume.
Once we have a notion of distance, we have a corresponding notion of
convergence.
Definition A.2 (Convergent and Cauchy Sequences).
Let
X
be a met
ric space with metric d
,
and let
{
f
n
}
n
∈
N
be a sequence of elements of
X.
(a) We say that
{
f
n
}
n
∈
N
converges
to
f
∈
X
if
lim
n
→∞
d(
f
n
, f
) = 0
,
i.e., if
∀
ε >
0
,
∃
N >
0
,
∀
n
≥
N,
d(
f
n
, f
)
< ε.
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 Fall '09
 HEIL
 Topology, Space, Metric space, Hilbert space, Topological space, Inner Products

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