FUNCTIONAL ANALYSIS
1
Douglas N. Arnold
2
References:
John B. Conway, A Course in Functional Analysis, 2nd Edition, SpringerVerlag, 1990.
Gert K. Pedersen, Analysis Now, SpringerVerlag, 1989.
Walter Rudin, Functional Analysis, 2nd Edition, McGraw Hill, 1991.
Robert J. Zimmer, Essential Results of Functional Analysis, University of Chicago Press,
1990.
CONTENTS
I. Vector spaces and their topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
FUNCTIONAL ANALYSIS
1
Douglas N. Arnold
2
References:John B. Conway,
A Course in Functional Analysis
, 2nd Edition, SpringerVerlag, 1990.Gert K. Pedersen,
Analysis Now
, SpringerVerlag, 1989.Walter Rudin,
Functional Analysis
, 2nd Edition, McGraw Hill, 1991.Robert J. Zimmer,
Essential Results of Functional Analysis
, University of Chicago Press,1990.
CONTENTS
I. Vector spaces and their topology
...............................................
2Subspaces and quotient spaces
............................................
4Basic properties of Hilbert spaces
.........................................
5II. Linear Operators and Functionals
..............................................
9The Hahn–Banach Theorem
..............................................
10Duality
..................................................................
10III. Fundamental Theorems
.......................................................
14The Open Mapping Theorem
.............................................
14The Uniform Boundedness Principle
......................................
15The Closed Range Theorem
..............................................
16IV. Weak Topologies
.............................................................
18The weak topology
.......................................................
18The weak* topology
......................................................
19V. Compact Operators and their Spectra
........................................
22Hilbert–Schmidt operators
...............................................
22Compact operators
.......................................................
23Spectral Theorem for compact selfadjoint operators
......................
26The spectrum of a general compact operator
.............................
28VI. Introduction to General Spectral Theory
......................................
31The spectrum and resolvent in a Banach algebra
.........................
31Spectral Theorem for bounded selfadjoint operators
......................
35
1
These lecture notes were prepared for the instructor’s personal use in teaching a halfsemester courseon functional analysis at the beginning graduate level at Penn State, in Spring 1997. They are certainlynot meant to replace a good text on the subject, such as those listed on this page.
2
Department of Mathematics, Penn State University, University Park, PA 16802.Email: dna@math.psu.edu. Web: http://www.math.psu.edu/dna/.1
2
I. Vector spaces and their topology
Basic deﬁnitions: (1) Norm and seminorm on vector spaces (real or complex). A normdeﬁnes a Hausdorﬀ topology on a vector space in which the algebraic operations are continuous, resulting in a
normed linear space
. If it is complete it is called a Banach space.(2) Inner product and semiinnerproduct. In the real case an inner product is a positivedeﬁnite, symmetric bilinear form on
X
×
X
→
R
. In the complex case it is positive deﬁnite,Hermitian symmetric, sesquilinear form
X
×
X
→
C
. An (semi) inner product gives riseto a (semi)norm. An inner product space is thus a special case of a normed linear space.A complete inner product space is a Hilbert space, a special case of a Banach space.The polarization identity expresses the norm of an inner product space in terms of theinner product. For real inner product spaces it is(
x,y
) =14(
x
+
y
2
−
x
−
y
2
)
.
For complex spaces it is(
x,y
) =14(
x
+
y
2
+
i
x
+
iy
2
−
x
−
y
2
−
i
x
−
iy
2
)
.
In inner product spaces we also have the parallelogram law:
x
+
y
2
+
x
−
y
2
= 2(
x
2
+
y
2
)
.
This gives a criterion for a normed space to be an inner product space. Any norm comingfrom an inner product satisﬁes the parallelogram law and, conversely, if a norm satisﬁes theparallelogram law, we can show (but not so easily) that the polarization identity deﬁnesan inner product, which gives rise to the norm.(3) A
topological vector space
is a vector space endowed with a Hausdorﬀ topology suchthat the algebraic operations are continuous. Note that we can extend the notion of Cauchysequence, and therefore of completeness, to a TVS: a sequence
x
n
in a TVS is Cauchy if for every neighborhood
U
of 0 there exists
N
such that
x
m
−
x
n
∈
U
for all
m,n
≥
N
.A normed linear space is a TVS, but there is another, more general operation involvingnorms which endows a vector space with a topology. Let
X
be a vector space and supposethat a family
{·
α
}
α
∈A
of seminorms on
X
is given which are
suﬃcient
in the sense that
α
{
x
α
= 0
}
= 0. Then the topology generated by the sets
{
x
α
< r
}
,
α
∈ A
,
r >
0,makes
X
a TVS. A sequence (or net)
x
n
converges to
x
iﬀ
x
n
−
x
α
→
0 for all
α
. Notethat, a fortiori,

x
n
α
−
x
α
→
0, showing that each seminorm is continuous.If the number of seminorms is ﬁnite, we may add them to get a norm generating thesame topology. If the number is countable, we may deﬁne a metric
d
(
x,y
) =
n
2
−
n
x
−
y
n
1 +
x
−
y
n
,
3
so the topology is metrizable.Examples: (0) On
R
n
or
C
n
we may put the
l
p
norm, 1
≤
p
≤ ∞
, or the weighted
l
p
norm with some arbitrary positive weight. All of these norms are equivalent (indeedall norms on a ﬁnite dimensional space are equivalent), and generate the same Banachtopology. Only for
p
= 2 is it a Hilbert space.(2) If Ω is a subset of
R
n
(or, more generally, any Hausdorﬀ space) we may deﬁne thespace
C
b
(Ω) of bounded continuous functions with the supremum norm. It is a Banachspace. If
X
is compact this is simply the space
C
(Ω) of continuous functions on Ω.(3) For simplicity, consider the unit interval, and deﬁne
C
n
([0
,
1]) and
C
n,α
([0
,
1]),
n
∈
N
,
α
∈
(0
,
1]. Both are Banach spaces with the natural norms.
C
0
,
1
is the space of Lipschitz functions.
C
([0
,
1])
⊂
C
0
,α
⊂
C
0
,β
⊂
C
1
([0
,
1]) if 0
< α
≤
β
≤
1.(4) For 1
≤
p <
∞
and Ω an open or closed subspace of
R
n
(or, more generally, a
σ
ﬁnitemeasure space), we have the space
L
p
(Ω) of equivalence classes of measurable
p
th powerintegrable functions (with equivalence being equality oﬀ a set of measure zero), and for
p
=
∞
equivalence classes of essentially bounded functions (bounded after modiﬁcationon a set of measure zero). For 1
< p <
∞
the triangle inequality is not obvious, it isMinkowski’s inequality. Since we modded out the functions with
L
p
seminorm zero, thisis a normed linear space, and the RieszFischer theorem asserts that it is a Banach space.
L
2
is a Hilbert space. If meas(Ω)
<
∞
, then
L
p
(Ω)
⊂
L
q
(Ω) if 1
≤
q
≤
p
≤∞
.(5) The sequence space
l
p
, 1
≤
p
≤ ∞
is an example of (4) in the case where themeasure space is
N
with the counting measure. Each is a Banach space.
l
2
is a Hilbertspace.
l
p
⊂
l
q
if 1
≤
p
≤
q
≤∞
(note the inequality is reversed from the previous example).The subspace
c
0
of sequences tending to 0 is a closed subspace of
l
∞
.(6) If Ω is an open set in
R
n
(or any Hausdorﬀ space), we can equip
C
(Ω) with thenorms
f
→ 
f
(
x
)

indexed by
x
∈
Ω. This makes it a TVS, with the topology being thatof pointwise convergence. It is not complete (pointwise limit of continuous functions maynot be continuous).(7) If Ω is an open set in
R
n
we can equip
C
(Ω) with the norms
f
→
f
L
∞
(
K
)
indexedby compact subsets of Ω, thus deﬁning the topology of uniform convergence on compactsubsets. We get the same toplogy by using only the countably many compact sets
K
n
=
{
x
∈
Ω :

x
≤
n,
dist(
x,∂
Ω)
≥
1
/n
}
.
The topology is complete.(8) In the previous example, in the case Ω is a region in
C
, and we take complexvalued functions, we may consider the subspace
H
(Ω) of holomorbarphic functions. ByWeierstrass’s theorem it is a closed subspace, hence itself a complete TVS.(9) If
f,g
∈
L
1
(
I
),
I
= (0
,
1) and
10
f
(
x
)
φ
(
x
)
dx
=
−
10
g
(
x
)
φ
(
x
)
dx,
4
for all inﬁnitely diﬀerentiable
φ
with support contained in
I
(so
φ
is identically zero near0 and 1), then we say that
f
is weakly diﬀerentiable and that
f
=
g
. We can then deﬁnethe
Sobolev space
W
1
p
(
I
) =
{
f
∈
L
p
(
I
) :
f
∈
L
p
(
I
)
}
, with the norm
f
W
1
p
(
I
)
=
10

f
(
x
)

p
dx
+
10

f
(
x
)

p
dx
1
/p
.
This is a larger space than
C
1
(¯
I
), but still incorporates ﬁrst order diﬀerentiability of
f
.The case
p
= 2 is particularly useful, because it allows us to deal with diﬀerentiabilityin a Hilbert space context. Sobolev spaces can be extended to measure any degree of diﬀerentiability (even fractional), and can be deﬁned on arbitrary domains in
R
n
.
Subspaces and quotient spaces.
If
X
is a vector space and
S
a subspace, we may deﬁne the vector space
X/S
of cosets.If
X
is normed, we may deﬁne
u
X/S
= inf
x
∈
u
x
X
, or equivalently
¯
x
X/S
= inf
s
∈
S
x
−
s
X
.
This is a seminorm, and is a norm iﬀ
S
is closed.
Theorem.
If
X
is a Banach space and
S
is a closed subspace then
S
is a Banach spaceand
X/S
is a Banach space.Sketch.
Suppose
x
n
is a sequence of elements of
X
for which the cosets ¯
x
n
are Cauchy.We can take a subsequence with
¯
x
n
−
¯
x
n
+1
X/S
≤
2
−
n
−
1
,
n
= 1
,
2
,...
. Set
s
1
= 0, deﬁne
s
2
∈
S
such that
x
1
−
(
x
2
+
s
2
)
X
≤
1
/
2, deﬁne
s
3
∈
S
such that
(
x
2
+
s
2
)
−
(
x
3
+
s
3
)
X
≤
1
/
4,
...
. Then
{
x
n
+
s
n
}
is Cauchy in
X ...
A converse is true as well (and easily proved).
Theorem.
If
X
is a normed linear space and
S
is a closed subspace such that
S
is a Banach space and
X/S
is a Banach space, then
X
is a Banach space.
Finite dimensional subspaces are always closed (they’re complete). More generally:
Theorem.
If
S
is a closed subspace of a Banach space and
V
is a ﬁnite dimensional subspace, then
S
+
V
is closed.Sketch.
We easily pass to the case
V
is onedimensional and
V
∩
S
= 0. We then have that
S
+
V
is algebraically a direct sum and it is enough to show that the projections
S
+
V
→
S
and
S
+
V
→
V
are continuous (since then a Cauchy sequence in
S
+
V
will lead to aCauchy sequence in each of the closed subspaces, and so to a convergent subsequence).Now the projection
π
:
X
→
X/S
restricts to a 11 map on
V
so an isomorphism of
V
ontoits image¯
V
. Let
µ
:¯
V
→
V
be the continuous inverse. Since
π
(
S
+
V
)
⊂
¯
V
, we may formthe composition
µ
◦
π

S
+
V
:
S
+
V
→
V
and it is continuous. But it is just the projectiononto
V
. The projection onto
S
is
id
−
µ
◦
π
, so it is also continuous.