Analytic Banach Space Valued Functions
Let B be a Banach space and D be an open subset of C.
Denition 1 (Analytic) Let f : D B.
(a) f is analytic at z0 D if
f (z0 ) = lim
z z0
f (z )f (z0 )
z z0
exists (in the norm on B).
(b) f is weakly analytic at z0 D
Another Riesz Representation Theorem
In these notes we prove (one version of) a theorem known as the Riesz Representation
Theorem. Some people also call it the RieszMarkov Theorem. It expresses positive linear
functionals on C (X ) as integrals over X . F
The Principle of Uniform Boundedness, and Friends
In these notes, unless otherwise stated, X and Y are Banach spaces and T : X Y
is linear and has domain X .
Theorem 1
(a) T is bounded if and only if
T 1 y Y
y
Y
1
=
xX
Tx
Y
1
has nonempty interior. (X , Y
Review of Hilbert and Banach Spaces
Denition 1
operations,
(Vector Space) A vector space over C is a set V equipped with two
(v , w ) V V v + w V
(, v) C V v V
called addition and scalar multiplication, respectively, that obey the following axioms.
Additi
Spectral Theory Examples
Example 1 (Spectrum of Multiplication Operators) Let
(X, M, ) be a nite (actually seminite will do) measure space,
1 p and
a : X C be a bounded measurable function on X .
Dene the bounded operator A : Lp (X, M, ) Lp (X, M, ) by
The Spectrum of Periodic Schrdinger Operators
o
I The Physical Basis for Periodic Schrdinger Operators
o
Let d IN and let 1 , , d be a set of d linearly independent vectors in IRd .
Construct a crystal by xing identical particles at the points of the latt
Completion
Theorem 1 (Completion) If V , , V is any inner product space, then there exists a Hilbert
space H, , , H and a map U : V H such that
(i) U is 11
(ii) U is linear
(iii) U x, U y H = x, y V for all x, y V
(iv) U (V ) = U x x V is dense in H. If V
Compact Operators
In these notes we provide an introduction to compact linear operators on Banach and
Hilbert spaces. These operators behave very much like familiar nite dimensional matrices,
without necessarily having nite rank. For more thorough treatme
Review of Integration
Denition 1 (Integral) Let (X, M, ) be a measure space and E M.
(a) L+ (X, M) =
f is Mmeasurable . If f L+ (X, M), then
f : X [0, ]
n
ai (Ei E )
f (x) d(x) = sup
n IN, 0 ai < , Ei M for all 1 i n
i=1
E
n
ai Ei (x) f (x) for all x X
an
Lattices and Periodic Functions
Denition L.1 Let f (x) be a function on IRd .
a) The vector IRd is said to be a period for f if
for all x IRd
f (x + ) = f (x)
b) Set
Pf =
IRd
is a period for f
If , Pf then + Pf and if Pf then Pf (sub x = z into f (x + )
Review of Unbounded Operators
Denition 1 Let H1 and H2 be Hilbert spaces and T : D (T ) H1 H2 be a linear
operator with domain D (T ).
(a) The graph of T is
(T ) =
( , T )
D (T )
H1 H2
(b) The operator T is said to be closed if (T ) is a closed subset o
An Elliptic Function The Weierstrass Function
Denition W.1 An elliptic function f (z ) is a non constant meromorphic function on C
that is doubly periodic. That is, there are two nonzero complex numbers 1 , 2 whose ratio
is not real, such that f (z + 1 )
Review of Spectral Theory
Denition 1 Let H be a Hilbert space and A L(H).
(a) A is called selfadjoint if A = A .
(b) A is called unitary if A A = AA = 1 Equivalently, A is unitary if it is bijective (i.e.
l.
11 and onto) and preserves inner products.
(c)
Review of Signed Measures and the RadonNikodym Theorem
Let X be a nonempty set and M P (X ) a algebra.
Denition 1 (Signed Measures)
(a) A signed measure on (X, M) is a function : M [, ] such that
(i) () = 0
(ii) assumes at most one of the values .
(iii) I
Review of Measurable Functions
Denition 1 (Measurable Functions) Let X and Y be nonempty sets and M and N
be algebras of subsets of X and Y respectively.
(a) A function f : X Y is said to be (M, N )measurable if
E N = f 1 (E )
xX
f (x) E
M
(b) A function
Review of Measure Theory
Let X be a nonempty set. We denote by P (X ) the set of all subsets of X .
Denition 1 (Algebras)
(a) An algebra is a nonempty collection A of subsets of X such that
i) A, B A = A B A
ii) A A = Ac = X \ A A
(b) A collection A of su