# Functional Analysis

## Period: Fall 2011

### Please note: in Spring 2012 Marcel de Jeu will organize a functional analysis seminar in Leiden on Friday afternoons. All participants in our course are welcome in this seminar!

Lecturers: Marcel de Jeu (Leiden University) and Andre Ran (Vrije Universiteit).
Assistants: Blaz Mramor (Vrije Universiteit) and Miek Messerschmidt (Leiden University).

Venue: Buys Ballot building, Utrecht University, room 061.
Time and dates of lectures: Tuesdays from 10.15-13.00 on September 6, 13, 20, 27, October 4, 11, 25, November 1, 8, 15, 22, 29, and December 6, 13 (i.e., all Tuesdays in the weeks 36 through 50, with a break in week 42).

Coordinates of Marcel de Jeu
Visiting address: Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), office 218 (third floor).
Telephone: 071 527 7118.
Email: mdejeu@math.leidenuniv.nl.
Mathematical Institute
Leiden University
P.O. Box 9512
2300 RA Leiden

Coordinates of Andre Ran
Visiting address: Vrije Universiteit, De Boelelaan 1081, office R3.45.
Telephone: 020 598 7691.
Email: acm.ran@few.vu.nl.
Afdeling Wiskunde
Vrije Universiteit
Faculteit der Exacte Wetenschappen
De Boelelaan 1081a
1081 HV Amsterdam

Coordinates of Blaz Mramor
Visiting address: Vrije Universiteit, De Boelelaan 1081.
Telephone: .
Email: bmramor@few.vu.nl.
Afdeling Wiskunde
Vrije Universiteit
Faculteit der Exacte Wetenschappen
De Boelelaan 1081a
1081 HV Amsterdam

Coordinates of Miek Messerschmidt
Visiting address: Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building). Email: miek@math.leidenuniv.nl
tel: 071 5277047
Mathematical Institute
Leiden University
P.O. Box 9512
2300 RA Leiden

The text we will use is the book by J.B. Conway: A course in Functional Analysis, second edition. Springer Verlag.

We assume some prior knowledge of Functional Analysis, although most of what we assume will be covered fairly quickly in the first three weeks. The goal of the course is to cover as much of the book as we can.

Examination
In principle exclusively on the basis of homework: an average grade of at least 5.5 for the homework assignments implies passing. However, although it is unlikely to occur, the lecturers reserve the right to invite individual students for an oral exam if necessary and base the final grade on both the homework and the oral exam.

Homework
The homework will be assigned at every even numbered lecture, with a deadline three workable weeks later. It can be handed in at the lecture, by email or by regular mail, the latter two preferably directly addressed to the assistant. If it is handed in between three weeks and four weeks later it is still accepted, but one point will be deducted from the grade for that particular assignment. Assignments will be accepted later than four weeks after the assignment only in case of clear farce majeure. The corrected homework will be handed back five weeks after the assignment, or sent back by regular mail in case of the assignments 6 and 7 (please write your postal address on the work).

Schedule and assignments:

Homework 1: assigned September 13, deadline October 4:

• I.1.11, I.2.4, I.3.3, I.4.10, I.4.11
• II.1.1, II.2.7
• III.1.8, III.1.16, III.4.6.

Homework 2: assigned September 27, deadline October 25. One week extra: no lecture on October 18.
To be handed in at a lecture, or with Miek Messerschmidt.

• III.12.7, III.13.4, III.14.6.
• IV.1.11, IV.1.12, IV.1.15.
• An exercise which is not in the book, and which tells you how to decide whether a given linear functional on a LCS is continuous:
"Prove the equivalence of (b) and (f) in Theorem IV.3.1."

Homework 3: assigned October 11, deadline November 1.
To be handed in at a lecture, or with Miek Messerschmidt.

• IV.3.5.
• Correctly formulated generalisation of IV.3.7: Let X be a TVS, and let f: X -> R be a non-zero (!) continuous R-linear functional on X. If A is a non-empty open convex subset of X, then f(A) is an open interval (possibly unbounded) on the real line. Prove this. (This was used in a proof in Lecture 5).
• V.1.7. You may simplify the exercise and prove the existence of such a sequence where y_n is a convex combination of finitely many of the x_i, not necessarily of the first n.
• Exercise not in the book: Let X be a compact Hausdorff space and suppose that the sequence f_1, f_2, ... of continuous functions on X is uniformly bounded (i.e., there exists M\geq 0 such that |f_n(x)|\leq M, for all n and x) and pointwise convergent to some continuous function f on X. Show that there is a sequence of convex linear combinations of the f_i that converges uniformly on X to f. (Hint: you will need the Riesz Representation Theorem in Appendix C).
• V.1.10. Somewhere along the line, you may need to prove that any weakly open neighbourhood of 0 contains a nonzero subspace. Weakly open sets in a infinite dimensional normed space are quite large!
• V.3.3. Now why would this exercise be in this section?
• Exercise related to section V.4: let X be a compact Hausdorff space. Prove that C(X) is reflexive if and only if X has only finitely many points.
This is probably the hardest exercise in this assignment. If C(X) is reflexive, think about V.4.2.d, A.2.6 and applying Urysohn's lemma very often around a fixed point in X, in order to show that the function which is equal to 1 in that point and zero elsewhere, is continuous.

Homework 4: assigned November 1, deadline November 22.
• VI.1.8, VI.3.10 with X=[0,1] and \mu the Lebsegue measure
• VII.1.1, You may assume in Exercise VII.1.1 that the maps (x,a) -> ax and (x,a) -> xa are continuous.
• VII.2.5 (only the first part),
• VII.3.5, VII.4.9, VII.6.5 and 6, VII.8.1

• VII.7.4.
• II.4.8 (h and g should both be in H, not K; the operator is assumed not to be the zero operator; the exercise discusses only BOUNDED rank one operators).
• II.5.4, II.6.4.
• XI 2.3.a, XI.3.3, XI.3.7.

Homework 6: assigned November 29, deadline January 3. Note that there is no "one-point-deduction-week" after these five weeks!!

• VIII.1.2.
• VIII.2.2.
• VIII.3.10 (Re A stands for the self-adjoint elements of A).
• VIII.3.25.
• VIII.4.7. It is obvious that I+J is a two-sided ideal: the point is that it is closed. Can you prove this? It is not trivial at all, but using the appropriate theorems the proof is just two or three lines.
• Exercise not in the book: prove Proposition 3.5 on page 240, i.e., show that a positive element has a unique positive n-th root, for each integer n≥1. You may assume that A has a unit element. Outline:
1. Use the commutative Gelfand-Naimark theorem for C*(1,a) to conclude that there exists at least one positive n-th root.
2. Show that any other n-th root, positive or not, commutes with the root from the first part.
3. Apply the commutative Gelfand-Naimark theorem to a suitably chosen C*-algebra to conclude the uniqueness of the positive root.
Homework 7: assigned December 13, deadline January 17. Note that there is no "one-point-deduction-week" after these five weeks!!
• VIII.5.11: You can assume that A has an identity element. As in the proof of "the" Gelfand-Naimark Theorem (5.17) as it was given in the lectures (the proof in the book is different), one only needs to show that \pi_f is injective, as a consequence of the second part of Theorem 4.8. Suppose, then, that \pi_f(a)=0 or, equivalently, that \pi_f(a*a)=0. Can you show that this implies that f_n(a*a)=0 for all n, and conclude from this that a*a=0? (Aside, the point of the exercise is to show that, for separable C*-algebras, one needs only ONE state for the Gelfand-Naimark theorem 5.17. A moment's thought shows that this representation is actually even separable).
• IX.1.7 and IX.1.8: both only for Example 1.6 on page 257, not for the other two. In this Example 1.6, the measure is assume to be finite.
• IX.1.14. For "only if", think of IX.2.2.b. For "if", it can be handy to realise that the representation extends to the bounded Borel functions, and that representations of C*-algebras are of course always positive.
• IX.2.9 Think of the continuous functional calculus, rather than the results in this section.
• IX.2.15. Here N is normal. You need not answer the question about the number of square roots.

Tentative programme (the numbers refer to the paragraphs in Conway's book)
Lecture 1 (Andre Ran, September 6: 1.1-1.6 and Appendix A.2
Lecture 2 (Andre Ran, September 13): 2.1-2.3 and 3.1-3.4
Lecture 3 (Andre Ran, September 20): 3.5-3.6, 3.10-3.14
Lecture 4 (Marcel de Jeu, September 27): 4.1-4.3
Lecture 5 (Marcel de Jeu, October 4): 5.1-5.4
Lecture 6 (Marcel de Jeu, October 11): 5.7-5.9
Lecture 7 (Andre Ran, October 25): 6.1, 6.3, 7.1-7.2
Lecture 8 (Andre Ran, November 1): 7.3-7.6, 7.8
Lecture 9 (Andre Ran, November 8): 7.7, 2.4-2.6
Lecture 10 (Andre Ran, November 15): bird's eye view of Chapter 11
Lecture 11 (Marcel de Jeu, November 22): 8.1-8.2
Lecture 12 (Marcel de Jeu, November 29): 8.3-8.5
Lecture 13 (Marcel de Jeu, December 6): 9.1-9.2
Lecture 14 (Marcel de Jeu, December 13): 9.4-9.5, definition of von Neumann algebras and basic results

Material actually covered

Lecture 1 (Andre Ran, September 6):
We covered Chapter 1 and Appendix A.2. Much of the material in Chapter 1 should be familiar from an introductory course, with the possible exception of working with a basis of a Hilbert space which is not necessarily countable. For such (non-separable) spaces it is relevant to give meaning to generalized series as in Lemma 4.12, and in order to do this we need the material on nets and their convergence in Appendix A.2. It is important to understand this material on nets, not just because we needed them for Hilbert spaces, but more generally since in non-metric spaces -- which occur naturally on many occasions -- sequences are no longer sufficient to work with, but nets are. As an example, Proposition A.2.4 is false when nets are replaced with sequences: it is always true that a continuous map transforms convergent sequences into convergent sequences, but this does not characterize continous maps in general.

Lecture 2 (André Ran, September 14, 2010):
This week we covered the largest part of Sections 1-3 from Chapter II, and the first three sections of Chapter III. Bounded linear operators were discussed on Hilbert spaces, with the important observation that boundedness and continuity are equivalent. Properties of the adjoint were discussed, and several classes of operators that are characterized by a certain relation between the operator and its adjoint were introduced (selfadjoint, normal, isometric, unitary). Projections and idempotents were introduced. A slight warning is in place: for Conway a projection is always an orthogonal projection. After this, Banach spaces were introduced. It is important that you try to understand all the examples given in the first section of chapter III. Linear operators were introduced, and finally, several propositions were discussed in which finite dimensionality plays a role somewhere.

Lecture 3 (André Ran, September 21, 2010):
In the third week we covered the most important theorems from the introductory courses: the Hahn-Banach theorem, the open mapping and closed graph theorems, the inverse mapping theorem and the uniform boundedness principle. All this is nicely explained in the book in sections 3.5, 3.6 and 3.10-14.

Lecture 4 (Marcel de Jeu, September 27, 201q):
We concentrated on IV.1 mostly, giving a thorough motivation of and introduction to locally convex spaces. Among the things to remember are how to introduce the topology in terms of a separating family of seminorms, what it means for a net to converge in the resulting topology, and Proposition IV.1.15 which validates the name "locally convex space".
To get some practice in arguing in a TVS, we also proved the equivalence of (b) and (d) in Theorem IV.3.1. Conway claims that his earlier proof for normed spaces works here again, but if you look at III.5.3 you will see that this is not so obvious. At that point he uses crucially that there is only one norm topology on a finite dimensional space. Now it is, in fact, also the case that there is only one topology which makes a finite dimensional space into a TVS (so that, in particular, each finite dimensional TVS is normable), but this is not proved in the book. So his proof is not entirely satisfactory, and since the characterization of continuous linear functionals as those linear functionals having a closed kernel is important later on, we have given an independent proof.
We also covered example IV.3.16. This is a topological vector space having only itself as a non-empty open convex subset, and consequently only 0 as a continuous functional. As we will see, this is very far from the locally convex case where continuous linear functionals exist in abundance. Our geometric intuition lets us down in this example, where each element of the space is a convex combination of elements of an arbitrarily small open metric ball centered at the origin.
Next time we will finish the paragraphs 1-3 of Chapter IV and make a dent in Chapter V.

Lecture 5 (Marcel de Jeu, October 4, 2011):
We finished Chapter IV.
In IV.2, Proposition 2.1 is the result most often used. Metrizable spaces are good to work in. If a TVS is metrizable and then complete as a metric spaces (these are the so-called Fréchet spaces), then the Baire Category Theorem holds and one can prove the consequences thereof as we already know them for Banach spaces, such as the Open Mapping Theorem, Closed Graph Theorem, Bounded Inverse Theorem and Uniform Boundedness Principle. These are all true for Fréchet spaces (but this is not proved in the book).
Section IV.3 starts with separation theorems for a TVS. Theorem 3.7 is then about as good as one can get, and the key transition to separation theorems for LCS is made in Theorem 3.9. Separation theorems imply the usual Hahn-Banach theorems as we know them from the normed case: after applying a separation theorem, one simply observes that a linear functional which maps a linear subspace into a half plane (or half line) in the field must actually be zero. This gives the basic HB-result 3.15 and (this is not in the book!) we used that one in turn to show that a continuous linear functional on a (not necessarily closed) linear subspace of a LCS extends to a continuous linear functional on the whole space. The moral of all this is, that in locally convex spaces one has a dual which is large enough so that the "usual" basic theorems about continuous linear functionals hold: local convexity gives Hahn-Banach type results.
Incidentally, some authors call a Fréchet spaces a space which is metrizable and then complete, as well as locally convex. It is now clear why such spaces are very good spaces to work in: one has the consequences of the category theorem at one's disposal, as well as a large dual to work with.
We made a start with Chapter V, by showing how a separating linear subspace V of the algebraic dual of a vector space X can be used to introduce a locally convex topology on X, such that the (continuous) dual of X is precisely the V one started with. Special cases are the weak topology on a LCS X and the weak*-topology on the dual X*. Theorem 1.2 and 1.3 simply reflect the general principle of the construction: the dual is the V one started with. We also showed that, if two locally convex topologies on X give the same dual, then the closure of a convex subset of X will be the same in both topologies. The items 1.4 and 1.5 follow from this general result. A (deeper) theorem of Mackey (not in the book) states that two locally convex topologies with the same dual also have the same bounded subsets (no convexity assumption on these sets).
Next time we will move forward to, amongst others, the Banach-Alaoglu theorem and the Krein-Milman theorem. These two combined are very a powerful tool to show existence of objects of a desired type in many situation.

Lecture 6 (Marcel de Jeu, October 11, 2011)
We concluded the mini series of three lectures on locally convex spaces by covering more material from Chapter V and giving an application to ergodic theory (which is neither in the book nor in he homework).
The first main result was the Bipolar Theorem 1.8, which together with 1.9 implies that the closed (=weakly closed) linear subspaces of a LCS X are in bijection with the weak*-closed linear subspaces in X*. The bijection is given by taking annihilators (these are the polars and prepolars of linear subspaces).
The second main result (we skipped V.2) was the Banach-Alaoglu theorem. We formulated the most general theorem, stating that the polar of an open neighborhood of 0 in a TVS X is always compact in the weak*-topology. This follows from Tychonov's theorem: see 3.15 in Rudin's FA book for the details of the proof. In Conway the result is stated and proved for normed spaces as V.3.1, and given as an exercise for locally convex spaces as exercise V.3.2, but it is actually true for topological vector spaces in general. The BA-theorem has consequences for reflexivity by relating it to weak compactness: see 4.2. In this context the (deep) Eberlein-Smulian theorem V.13.1 was mentioned, which states that for the weak topology on a normed space the notions of compactness and sequential compactness of a subset of X are the same. The same holds for metric spaces, but for general topological spaces the notions are independent: there are spaces which are compact but not sequentially compact, and there are spaces which are sequentially compact, but not compact.
We noted the metrizability result 5.1 and mentioned that, for a Banach space X, the metrizability of X in the weak topology and the metrizability of X* in the weak*-topology are both equivalent to the space being finite dimensional. This is not entirely trivial; see 2.5.14 and 2.6.12 in Megginson's "An introduction to Banach space theory" for a proof.
The third main result was the Krein-Milman theorem: 7.4+7.8. The Banach-Alaoglu theorem and the Krein-Milman theorem often occur in a highly powerful combination to show that certain "desired" objects exist, by first noting that these desired objects correspond to extremal points of a non-empty convex set in the dual of a LCS, next noting (Banach-Alaoglu) that this set is weak*-compact, and finally applying the Krein-Milman theorem to conclude that there are extremal points as needed. We gave an example of such a concrete application in ergodic theory, by showing that for any continous map from a compact Hausdorff space to itself there is always an ergodic probability measure.
In two week's time André Ran will start a new mini series of four lectures.

Lecture 7 (André Ran, October 25, 2011):
We started with Chapter VI, Sections 1 and 3. Section 1 on the adjoint of an operator acting between Banach spaces was covered up to and including Proposition 1.9. Next we introduced compact operators and completely continuous operators, and showed what the relation between the two concepts are (Proposition 3.3). We showed that if A is compact then its adjoint is compact. Also we showed that the set of compact operators is an ideal in the set of all bounded linear operators. Further, we discussed whether or not the closure of the set of operators with finite dimensional range is equal to the set of compact operators. Here we proved nothing, but it is pointed out that the discussion on pages 175 and 176 is very interesting.
The next topic was an introduction to Banach algebras, Sections VII.1 and 2. Examples were given, and we discussed the notions of ideals, quotients and maximal ideals. An important point is that the set of invertible elements in a Banach algebra is open.

Lecture 8 (André Ran, November 1, 2011):
This week the spectrum of an element in a Banach algebra was the main topic. We covered the Riesz functional calculus, the spectral mapping theorem, and we specialized to the Banach algebra of bounded linear operators on a Banach space. In particular, we showed that the boundary of the spectrum consists of approximate point spectrum.

Lecture 9 (André Ran, November 8, 2011):
We discussed the spectral properties of compact operators on Hilbert spaces. Contrary to the general Banach space situation, here we have that every compact operator is the limit in norm of a sequence of finite rank operators. We showed that every non-zero point in the approximate point spectrum of a compact operator T is actually an eigenvalue, and that for every non-zero eigenvalue the corresponding eigenspace is finite dimensional. Next, it was shown that if a non-zero complex number z is not an eigenvalue of T, and if the complex conjugate of z is not an eigenvalue of the adjoint of T, then z is not in the spectrum of T. It was then shown that the second condition is superfluous. Finally, the spectral theorem for compact selfadjoint operators was discussed.

Lecture 10 (André Ran, November 15, 2011):
The first highlight of this lecture was Section VII.8 on abelian Banach algebras, where we showed that every abelian Banach algebra A "is" a subalgebra of the set of continuous functions on a compact Hausdorff space X. The set X is actually the maximal ideal space, that is, the set of non-zero homomorphism from A to the complex numbers, equipped with the weak-star topology.
We then did a very brief introduction to Fredholm operators (chapter XI, only the basics of Sections 1,2,3). It was shown that Conway's definition of Fredholm operators is equivalent to the more usual one that states that A is Fredholm if and only if ker A is finite dimensional, ran A isclosed and ker A* is finite dimensional. We proved the logarithmic property of the Fredholm index. Finally we discussed two perturbation results, stating that the index is constant if one perturbs a Fredholm operator with either a compact or a small bounded operator.
Finally, we discussed Section II.6 on Sturm Liouville operators as an application of the spectral theorem of compact selfadjoint operators on a Hilbert space.

Lecture 11 (Marcel de Jeu, November 22, 2011):
We started working on C*-algebras and covered Sections 1 and 2 of Chapter VIII. The highlights are the automatic continuity of *-homomorphisms (Prop. 1.11 (d); this will be improved later on when we show that a non-zero *-homomorphism has norm 1 and a C*-subalgebra as image), Proposition 1.14 (Spectral permanence: the spectrum does not depend on the algebra, except perhaps the complex number zero which in the non-unital case may appear or disappear), the commutative Gelfand Naimark Theorem 2.1 (the commutative C*-algebras are, via the Gelfand transform, precisely the algebras C_0(X) with X locally compact Hausdorff) and the continuous functional calculus for normal elements (Theorem 2.6 and 2.7). The definition of the continuous functional calculus looks a bit involved at first sight, but if you keep the homeomorphism in Proposition 2.3 and the commutative Gelfand Naimark theorem in mind, it is completely obvious that the isomorphism which defines the continuous functional calculus does in fact exist, and that the unital C*-subalgebra generated by a normal element is isometrically *-isomorphic to the continuous functions on the spectrum of that element.

Lecture 12 (Marcel de Jeu, November 29, 2011):
We continued our study of basic properties of C*-algebras in Sections 3, 4 and 5 of Chapter VIII. Most of these ultimately rely on the commutative Gelfand-Naimark theorem (2.1) which provides, amongst others, a powerful functional calculus.
In Section 3 we concentrated on proving Proposition 3.7: the positive elements are a closed convex cone. This gives a partial ordering on the self-adjoint elements of A, which then becomes a partially ordered real vector space. Theorem 3.8 is important, because it shows that the two notions of positivity in B(H) coincide. Although not without interest, we did not cover the decomposition results 3.4 and 3.5 for reasons of time. The highlights in Section 4 are 4.3 (closed two-sided ideals are self-adjoint), 4.6 (the quotient norm has the C*-property) and 4.8 (the image of a *-homomorphism is a C*-subalgebra). Although not mentioned explicitly in the book, it then also follows immediately that A/Ker \rho is isomorphic to \rho (A) as C*-algebras, i.e., the usual isomorphism theorem holds in the strongest possible sense. In addition, it is also true that a non-zero *-homomorphism has norm 1. These results are all easily formulated and remembered, but they required quite some work to prove!
We made a dent in Section 5, by defining cyclic representations and observing that these are the building blocks of all representations (Theorem 5.9). The subsequent step, in the next lecture, will be on the remaining material in Section 5 (from 5.10 onward). We will see how there is a natural bijection between positive functionals on A and unitary equivalence classes of cyclic representations with a cyclic vector given (GNS-theorem 5.14), and also that there are sufficiently many positive functionals to build an injective (hence: isometric) representation of A (5.17, this is the famous Gelfand-Naimark theorem which states that the closed *-invariant subalgebras of B(H) are the only C*-algebras, up to isomorphism). After that, we continue in Chapter IX.

Lecture 13 (Marcel de Jeu, December 6, 2011):
We covered Section 5 of Chapter VIII. The GNS-theorem 5.14, slightly reformulated, shows that there is a bijection between positive functionals and unitary equivalence classes of cyclic representations with a given cyclic vector. So, to construct representations of a C*-algebra "all" one needs are positive functionals. These can be characterised as the continous functionals which assume their norm at the identity element (this precise version was proved during the lecture). This characterisation implies that any Hahn-Banach extension of a positive functional from a C*-subalgebra to the whole algebra is automatically positive, and since it is easy to find positive functionals on commutative C*-subalgebras (these are the positive measures on the spectrum) one has a rich supply of positive functionals, hence of representations. With all this at one's disposal, the proof of the Gelfand-Naimark theorem 5.17 is then easy: one rapidly concludes that there is an injective *-homomorphism into some B(H), for some large Hilbert space H, and from the general theory we then already know that this must be an isometric *-embedding. Hence, after the fact, any C*-algebras "is" a C*-algebra of operators and in retrospect this puts some of the earlier results, which were obvious for operators, into perspective.

Lecture 14 (Marcel de Jeu, December 13, 2011):
We gave a detailed demonstration of the main results of Section 1 in chapter IX, namely Proposition 1.12 and Theorem 1.14. Our proof was a bit different from Conway's, as we avoided invoking Proposition V.4.1 (Goldstine's Theorem) in the proof of Theorem 1.14 and also the notion of the weak operator topology. The key tools in both approaches, however, remains the Riesz Representation theorem describing the dual of C(X), which yields the complex valued measures in the proofs, and the well-known bijection between the bounded sesquilinear forms on a Hilbert space and the bounded operators on that space, which enables one to define operators from the measures.
The moral of Proposition 1.12 and Theorem 1.14, taken together, is that there is a natural bijection between representations of C(X) and regular spectral measures on the Borel sets of the compact Hausdorff space X, and that a representation of C(X) can be extended to a representation of the bounded Borel functions on X (using Proposition 1.12 again).
After the work done in Section 1, the version (there are many) of the spectral theorem which is given as Theorem 2.2 is an easy consequence; this concluded our work on spectral theory. One of the ways to remember the spectral theorem is that the continuous functional calculus can be extended to a Borel functional calculus, i.e., to a representation of the bounded Borel functions on the spectrum of N. The orthogonal projections of the spectral measure are then simply the images of the characteristic functions under this extended representation. It is then, e.g., obvious that N can be approximated by linear combinations of mutually orthogonal projections: simply approximate the function z uniformly on the spectrum by step functions. Since the extended representation is contractive, the images of the step functions (which are the linear combinations of projections as required) will approximate the image of the function z (i.e., approximate N) in the operator norm.
It should be obvious from these two sections that spectral theory and measure theory are intimately connected. This becomes even more obvious in the remaining sections in this chaper (which we did not cover), culminating in the Theorems 10.20 and 10.21, where normal operators on a separable Hilbert space are classified up to unitary equivalence. Multiplication operators on an L_2-space are obvious examples of normal operators and part of the statement of Theorems 10.20 and 10.21 is that every normal operator on a Hilbert space is built up from building blocks of this type.