Functional Analysis 
Feng Tian, and Palle Jorgensen 
Department of Mathematics 
14 MLH 
The University of Iowa 
Iowa City, IA 522421419 
USA. 
Notes from a course taught by Palle Jorgensen in the fall semester of 2009. The course covered central themes in functional analysis and operator theory, with an emphasis on topics of special relevance to such applications as representation theory, harmonic analysis, mathematical physics, and stochastic integration.
These are the lecture notes I took from a topic course taught by Professor Jorgensen during the fall semester of 2009. The course started with elementary Hilbert space theory, and moved very fast to spectral theory, completely positive maps, KadisonSinger conjecture, induced representations, selfadjoint extensions of operators, etc. It contains a lot of motivations and illuminating examples.
I would like to thank Professor Jorgensen for teaching such a wonderful course. I hope students in other areas of mathematics would benefit from these lecture notes as well.
Unfortunately, I have not been able to fill in all the details. The
notes are undergoing editing. I take full responsibility for any errors
and missing parts.
Feng Tian
03. 2010
Contents
 1 Elementary Facts

2 GNS, Representations
 2.1 Representations, GNS, primer of multiplicity
 2.2 States, dual and predual
 2.3 Examples of representations, proof of GNS
 2.4 GNS, spectral thoery
 2.5 Choquet, KreinMilman, decomposition of states
 2.6 Beginning of multiplicity
 2.7 Completely positive maps
 2.8 Comments on Stinespring’s theorem
 2.9 More on the CP maps
 2.10 KrienMilman revisited
 2.11 States and representation
 2.12 Normal states
 2.13 KadisonSinger conjecture
 3 Appliations to Groups
 4 Unbounded Operators
Chapter 1 Elementary Facts
1.1. Transfinite induction
Let be a paritially ordered set. A sebset of is said to be a chain, or totally ordered, if in implies that either or . Zorn’s lemma says that if every chain has a majorant then there exists a maximal element in .
Theorem 1.1.
(Zorn) Let be a paritially ordered set. If every chain in has a majorant (or upper bound), then there exists an element in so that implies , for all in .
An illuminating example of a partially ordered set is the binary tree model. Another example is when is a family of subsets of a given set, partially ordered by inclusion. Zorn’s lemma lies at the foundation of set theory. It is in fact an axiom and is equivalent to the axiom of choice and Hausdorff’s maximality principle.
Theorem 1.2.
(Hausdorff Maximality Principle) Let be a paritially ordered set, then there exists a maximal totally ordered subset in .
The axiom of choice is equivalent to the following statement on infinite product, which itself is extensively used in functional analysis.
Theorem 1.3.
(axiom of choice) Let be a family of nonempty sets indexed by . Then the infinite Cartesian product is nonempty.
can be seen as the set of functions from to . The point of using the axiom of choice is that if the index set is uncountable, there is no way to verify whether is in or not. It is just impossible to check for each that in contained in , for some coordinates will be unchecked. The power of transfinite induction is that it applies to uncountable sets as well. In case the set is countable, we simply apply the down to earth standard induction. The standard mathematical induction is equivalent to the Peano’s axiom which states that every nonempty subset of of the set of natural number has a unique smallest element.
The key idea in applications of the transfinite induction is to cook up in a clear way a partially ordered set, so that the maximum element turns out to be the object to be constructed. Examples include HahnBanach extension theorem, KreinMillman’s theorem on compact convex set, existance of orthonoral basis in Hilbert space, Tychnoff’s theorem on infinite Cartesian product of compact spaces, where the infinite product space is nonempty follows imediately from the axiom of choice.
Theorem 1.4.
(Tychonoff) Let be a family of compact sets indexed by . Then the infinite Cartesian product in compact with respect to the product topology.
We will apply the transfinite induction to show that every infinite dimensional Hilbert space has an orthonormal basis (ONB).
Classical functional analysis roughly divides into two branches

study of function spaces (Banach space, Hilbert space)

applications in physics and engineering
Within pure mathematics, it is manifested in

representation theory of groups and algebras

algebras, Von Neumann algebras

wavelets theory

harmonic analysis

analytic number theory
Definition 1.5.
Let be a vector space over . is a norm on if for all in and in


; implies

is a Banach space if it is complete with respect to the metric induced by .
Definition 1.6.
Let be vector space over . An inner product is a function so that for all in and in ,

is linear (linearity)

(conjugation)

; and implies (positivity)
In that case defines a norm on and is denote by . is said to be an inner product space is an inner product is defined. A Hilbert space is a complete inner product space.
Remark 1.7.
The abstract formulation of Hilbert was invented by Von Neumann in 1925. It fits precisely with the axioms of quantum mechanics (spectral lines, etc.) A few years before Von Neumann’s formulation, Heisenberg translated Max Born’s quantum mechanics into mathematics.
For any inner product space , observe that the matrix
is positive definite by the positivity axiom of the definition of an inner product. Hence the matrix has positive determinant, which gives rises to the famous CauchySchwartz inequality
An extremely useful way to construct a Hilbert space is the GNS construction, which starts with a semipositive definite funciton defined on a set . is said to be semipositive definite, if for finite collection of complex nubmers ,
Let be the span of where , and define a sesiquilinear form on as
However, the positivity condition may not be satisfied. Hence one has to pass to a quotient space by letting , and be the quotient space . The fact that is really a subspace follows from the CauchySchwartz inequality above. Therefore, is an inner product on . Finally, let be the completion of under and is a Hilbert space.
Definition 1.8.
Let be a Hilbert space. A family of vectors in is said to be an orthonormal basis of if

and

.
We are ready to prove the existance of an orthonormal basis of a Hilbert space, using transfinite induction. Again, the key idea is to cook up a partially ordered set satisfying all the requirments in the transfinite induction, so that the maximum elements turns out to be an orthonormal basis. Notice that all we have at hands are the abstract axioms of a Hilbert space, and nothing else. Everything will be developed out of these axioms.
Theorem 1.9.
Every Hilbert space has an orthonormal basis.
To start out, we need the following lemmas.
Lemma 1.10.
Let be a Hilbert space and . Then the following are equivalent:

implies

Lemma 1.11.
(GramSchmidt) Let be a sequence of linearly independent vectors in then there exists a sequence of unit vectors so that .
Remark.
The GramSchmidt orthogonalization process was developed a little earlier than Von Neumann’s formuation of abstract Hilbert space.
Proof.
we now prove theorem (1.9). If is empty then we are finished. Otherwise, let . If , we may consider which is a normalized vector. Hence we may assume . If we are finished again, otherwise there exists . By lemma (1.11), we may assume and . By induction, we get a collection of orthonormal vectors in .
Consider partially order by set inclusion. Let be a chain and let . is clearly a majorant of . We claim that is in the partially ordered system. In fact, for all there exist and in so that and . Since is a chain, we may assume . Hence and , which shows that is in the partially ordered system.
By Zorn’s lemma, there exists a maximum element . It suffices to show that the closed span of is . Suppose this is false, then by lemma (1.10) there exists so that . Since and is maximal, it follows that , which implies . By the positivity axiom of the definition of Hilbert space, .∎
Corollary 1.12.
Let be a Hilbert space, then is isomorphic to the space of the index set of an ONB of .
Remark 1.13.
There seems to be just one Hilbert space, which is true in terms of the Hilbert space structure. But this is misleading, because numerous interesting realizations of an abstract Hilbert space come in when we make a choice of the ONB. The question as to which Hilbert space to use is equivalent to a good choice of an ONB. This is simiar to the argument that there is just one set for each given cardinality in terms of set structure, but there are numerous choices of elements in the sets making questions interesting.
Suppose is separable, for instance let . Then . It follows that potentially we could choose a doublely indexed basis for . It turns out that this is precisely the setting of wavelet basis! What’s even better is that in space, there are all kinds of diagonalized operators, which correspond to selfadjoint (or normal) operators in . Among these operators in , we single out the scaling () and translation () operators, which are diagonalized, NOT simultaneously though.
1.1.1. path space measures
Let be the infinite Cartesian product of with the product topology. is compact and Hausdorff by Tychnoff’s theorem.
For each , let be the coordinate projection, and assign probability measures on so that and , where . The collection of measures satisfies the consistency conditiond, i.e. is the restriction of onto the coordinate space. By Kolomogorov’s extension theorem, there exists a unique probability measure on so that the restriction of to the coordinate is equal to .
It follows that is a sequence of independent identically distributed (i.i.d.) random variables in with and ; and .
Let be a separable Hilbert space with an orthonormal basis . The map extends linearly to an isometric embedding of into . Moreover, let be the symmetric Fock space. is the closed span of the the algebraic tensors , thus extends to an isomorphism from to .
1.2. Dirac’s notation
P.A.M Dirac was every efficient with notations, and he introduced the “braket” vectors. Let be a Hilbert space with inner product . We denote by “bra” for vectors and “ket” for vectors where .
With Dirac’s notation, our first observation is the followsing lemma.
Lemma 1.14.
Let be a unit vector. The operator can be written as . is a rankone selfadjoint projection.
Proof.
. Since
so . ∎
More generally, any rankone operator can be wrritten as sending to . With the braket notation, it’s easy to verify that the set of rankone operators forms an algebra, which easily follows from the fact that . The moment that an orthonormal basis is selected, the algebra of operators on will be translated to the algebra of matrices (infinite). Every Hilbert space has an ONB, but it does not mean in pratice it is easy to select one that works well for a particular problem.
It’s also easy to see that the operator
where is a finite set of orthonormal vectors in , is a selfadjoint projection. This follows, since
and .
The GramSchmidt orthogonalization process may now be written in Dirac’s notation so that the induction step is really just
which is a unit vector and orthogonal to . Notice that if is non separable, the standard induction does not work, and the transfinite induction is needed.
1.2.1. connection to quamtum mechanics
Quamtum mechanics was born during the years from 1900 to 1913. It was created to explain phenomena in black body radiation, hydrogen atom, where a discrete pattern occurs in the frequences of waves in the radiation. The radiation energy , with being the Plank’s constant. Classical mechanics runs into trouble.
During the years of 1925~1926, Heisenberg found a way to represent the energy as a matrix, so that the matrix entries represents the transition probability from energy to energy . A foundamental relation in quantum mechenics is the commutation relation satisfied by the momentum operator and the position operator , where
Heisernberg represented the operators by matrices, although his solution is not real matrices. The reason is for matrices, there is a trace operation where . This implies the trace on the lefthandside is zero, while the trace on the righhandside is not. This suggests that there is no finite dimensional solution to the commutation relation above, and one is forced to work with infinite dimensional Hilbert space and operators on it. Notice also that do not commute, and the above commutation relation leads to the uncertainty principle (Hilbert, Max Born, Von Neumann worked out the mathematics), which says that the statistical variance and satisfy . We will come back to this later.
However, Heisenberg found his “matrix” solutions, where
and
the complex in front of is to make it selfadjoint.
A selection of ONB makes a connection to the algebra operators acting on and infinite matrices. We check that using Dirac’s notation, the algebra of operators really becomes the algebr of infinite matrices.
Pick an ONB in , . We denote by the matrix of under the ONB. We compute .
where .
Let be a unit vector in . represents a quantum state. Since , the numbers represent a probability distribution over the index set. If and are two states, then
which has the interpretation so that the transition from to may go through all possible intermediate states . Two states are uncorrelated if and only if they are orthogonal.
1.3. Operators in Hilbert space
Definition 1.15.
Let be a linear operator on a Hilbert space .

is selfadjoint if

is normal if

is unitary if

is a selfadjoint projection if
Let be an operator, then we have , which are both selfadjoint, and . This is similar the to decomposition of a complex nubmer into its real and imaginary parts. Notice also that is normal if and only if and commute. Thus the study of a family of normal operators is equivalent to the study of a family of commuting selfadjoint operators.
Lemma 1.16.
Let be a complex number, and be a selfadjoint projection. Then is unitary if and only if .
Proof.
Since is a selfadjoint projection,
If then and is unitary. Conversely, if then it follows that . If we assume that is nondegenerate, then .∎
Definition 1.17.
Let be a linear operator on a Hilbert space . The resolvent is defined as
and the spectrum of is the complement of , and it is denoted by or .
Definition 1.18.
Let be the Borel algebra of . is a Hilbert space. is a projectionvalued measure, if

,


, if . The convergence is in terms of the strong operator topology.
Von Neumann’s spectral theorem states that an operator is normal if and only if there exits a projectionvalued measure on so that , i.e. is represented as an integral again the projectionvalued measure over its spectrum.
In quamtum mechanics, an observable is represented by a selfadjoint operator. Functions of observables are again observables. This is reflected in the spectral theorem as the functional calculus, where we may define using the spectral representation of .
The stardard diagonalization of Hermitian matrix in linear algebra is a special case of the spectral theorem. Recall that if is a Hermitian matrix, then where are the eigenvalues of and are the selfadjoint projections onto the eigenspace associated with . The projectionvalued measure in this case can be written as , i.e. the counting measure supported on .
Hersenberg’s commutation relation is an important example of two noncommuting selfadjoint operators. When is a selfadjoint projection acting on a Hilbert space , is a real number and it represents observation of the observable prepared in the state . Quantum mechanics is stated using an abstract Hilbert space as the state space. In practice, one has freedom to choose exactly which Hilbert space to use for a particular problem. The physics remains to same when choosing diffenent realizations of a Hilbert space. The concept needed here is unitary equivalence.
Suppose is a unitary operator, is a selfadjoint projection. Then is a selfadjoint projection on . In fact, where we used , as is unitary. Let be a state in and be the corresponding state in . Then
i.e. the observable has the same expectation value. Since every selfadjoint operator is, by the spectral theorem, decomposed into selfadjoint projections, it follows the expectation value of any observable remains unchanged under unitary transformation.
Definition 1.19.
Let and be operators. is unitarily equivalent to is there exists a unitary operator so that .
Example 1.20.
Fourier transform ,
The operators and are both densely defined on the Schwartz space . and are unitary equivalent via the Fourier transform,
Apply GramSchmidt orthogonalization to polynomials against the measrue , and get orthognoal polynomials. There are the Hermite polynomials (Hermit functions).
The Hermite functions form an orthonormal basis (normalize it) and transform and to Heisenberg’s infinite matrices. Some related operators: . It can be shown that
or equivalently,
. is called the energy operator in quantum mechanics. This explains mathematically why the energy levels are discrete, being a multiple of .
A multiplication operator version is also available which works especially well in physics. It says that is a normal operator in if and only if is unitarily equivalent to the operator of multiplication by a measurable function on where is compact and Hausdorff. We will see how the two versions of the spectral theorem are related after first introduing the concept of transformation of measure.
1.3.1. Transformation of measure
Let and be two measurable spaces with algebras and respectively. Let be a measurable function. Suppose there is a measure on . Then defines a measure on . is the transformation measure of under .
Notice that if , then if and only if . Hence
It follows that for simple function , and
With a standard approximation of measurable functions by simple functions, we have for any measurable function ,
The above equation is a generalization of the substitution formula in calculus.
The multiplication version of the spectral theory states that every normal operator is unitarily equivalent to the operator of multiplication by a measurable function on where is compact and Hausdorff. With transformation of measure, we can go one step further and get that is unitarily equivalent to the operator of multiplication by the independent variable on some space. Notice that if is nesty, even if is a nice measure (say the Lebesgue measure), the transformation meaure can still be nesty, it could even be singular.
Let’s assume we have a normal operator given by multiplication by a measurable function . Define an operator by
is unitary, since
Claim also that
To see this, let be a measurable function. Then
Recall we have stated two versions of the spectral theorem. (multiplication operator and projectionvalued measure) Consider the simplest case for the projectionvalued measure, where we work with . Claim that , i.e. the operator of multiplication by on the Hilbert space , is a projectionvalued measure.
Apply this idea to and , the momemtum and positon operators in quantum mechanics. , . As we discussed before,
in other words, and are unitarily equivalent via the Fourier transform, which diagonalizs . Now we get a projectionvalued measure (PVM) for by
This can be seen as the convolution operator with respect to the inverse Fourier transform of .
1.4. Lattice structure of projections
We first show some examples of using GramSchmidt orthoganoliztion to obtain orthonormal bases for a Hilbert space.
Example 1.21.
. The polynomials are linearly independent in , since if
then as an analytic function, the lefthandside must be identically zero. By StoneWeierstrass theorem, is dense in under the norm. Since , it follows that is also dense in . By GramSchmidt, we get a sequence of finite dimensional subspaces in , where has an orthonormal basis , so that . Define
The set is dense in , since and the latter is dense in . Therefore, forms an orthonormal basis of .
Example 1.22.
. Consider the set of complex exponentials . This is already an ONB for and leads to Fourier series. Equivalently, may also consider .
The next example constructs the Haar wavelet.
Example 1.23.
. Let be the characteristic function of . Define and . For fixed and , since they have disjoint support.
Exercise 1.24.
Let be the operator of multiplication by . Compute the matrix of under wavelet basis. (this is taken from Joel Anderson, who showed the implies that where is a diagonal operator and is a compact perturbation.d)
Theorem 1.25.
Let be a Hilbert space. There is a onetoone correspondence between selfadjoint projections and closed subspaces of .
Proof.
Let be a selfadjoint projection in . i.e. . Then is a closed subspace. Denote by the completement of , i.e. . Then . Since , therefore .
Conversely, let be a closed subspace in . First notice that the parallelogram law is satisfied in a Hilbert space, where for any , . Let , define . By definition, there exists a sequence in so that as . Apply the parallelogram law to and ,
which simplies to
Notice here all we require is lying in the subspace , hence it suffices to require simply that is a convex subset in . see Rudin or Nelson page 62 for more details. ∎
Von Neumann invented the abstract Hilbert space in 1928 as shown in one of the earliest papers. He work was greatly motivated by quantum mechanics. In order to express quantum mechanics logic operations, he created lattices of projections, so that everything we do in set theory with set operation has a counterpart in the operations of projections.
SETS  CHAR  PROJECTIONS  DEFINITIONS 
. This is similar to set operation where . In general, product and sum of projections are not projections. But if then the product is in fact a projection. Taking adjoint, one get . It follows that . i.e. containment implies the two projections commute.
During the same time period as Von Neumann developed his Hilbert space theory, Lesbegue developed his integration theory which extends the classical Riemann integral. The motone sequence of sets in Lebesgue’s integration theory also has a counterpart in the theory of Hilbert space. To see what happens here, let and we show that this implies .
Lemma 1.26.
.
Proof.
It follows from
∎
As a consequence, implies forms a monotone increasing sequence in , and the sequence is bounded by , since for all . Therefore the sequence converges to , in symbols
in the sense that (strongly convergent) for all , there exists a vector, which we denote by so that
and really defines a selfadjoint projection.
The examples using GramSchmidt can now be formulated in the lattice of projections. We have the dimensional subspaces and the orthogonal projection onto , where
Since is dense in , it follows that and . We may express this in the lattice notations by
The tensor product construction fits with composite system in quamtum mechanics.
Lemma 1.27.
. Then if and only if . i.e. .
Proof.
Notice that
If then , hence is a projection. Conversely, if , then implies that . Since it follows that hence . ∎
In terms of characteristic functions,
hence is a characteristic function if and only if .
The set of projections in a Hilbert space is partially ordered according to the corresponding closed subspaces paritially ordered by inclusion. Since containment implies commuting, the chain of projections is a family of commuting selfadjoint operators. By the spectral theorem, may be simultaneously diagonalized, so that is unitarily equivalent to the operator of multiplication by on the Hilbert space , where is a compact and Hausdorff space. Therefore the lattice structure of prjections in is precisely the lattice structure of , or equivalently, the lattice structure of measurable sets in .
Lemma 1.28.
Consider . The followsing are equivalent.

;

;

, for any ;

in the sense that , for any .
Proof.
The proof is trivial. Notice that
where we used that fact that
∎
1.5. Ideas in the spectral theorem
We show some main ideas in the spectral theorem. Since every normal operator can be written as where are commuting selfadjoint operators, the presentation will be focused on selfadjoint operators.
Let be a selfadjoint operator acting on a Hilbert space . There are two versions of the spectral theorem. The projectionvalued measure (PVM), and the multiplication operator .
1.5.1. Multiplication by

In this version of the spectral theorem, implies that is unitarily equivalent to the operator of multiplication by a measurable function on the Hilbert space , where is a compact Hausdorff space, and is a regular Borel measure.
induces a Borel measure on , supported on . Define the operator where
Then,
hence is unitary. Moreover,
it follows that is unitarily equivalent to on , and . The combined transformation diagonalizes as
It is seens as a vast extention of diagonalizing hermitian matrix in linear algebra, or a generalization of Fourier transform.

What’s involved are two algebras: the algebra of measurable functions on , treated as multiplication operators, and the algebra of operators generated by (with identity). The two algebras are isomorphic. The spectral theorem allows to represent the algebra of by the algebra of functions (in this direction, it helps to understand ); also represent the algebra of functions by algebra of operators generated by (in this direction, it reveals properties of the function algebra and the underlying space . We will see this in a minute.)

Let be the algebra of functions. is a representation, where
and we may define operator
This is called the spectral representation. In particular, the spectral theorem of implies the following substitution rule
is welldefined, and it extends to all bounded measurable functions.
Remark 1.29.
Notice that the map
is an algebra isomorphism. To check this,
where we used the fact that