Abstract: This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington, Dr. David Salinger, and Prof. Alex Strohmaier used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more.

The notes are available also for download in PDF.

The suggested textbooks are [, , , ]. The other nice books with many interesting problems are [, ].

Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them!

Contents

• 0 Motivating Example: Fourier Series
  • 0.1  Fourier series: basic notions
    • 0.1.1 2π-periodic functions
    • 0.1.2 Integrating the complex exponential function
  • 0.2 The vibrating string
    • 0.2.1 Separation of variables
    • 0.2.2 Principle of Superposition
  • 0.3 Historic: Joseph Fourier
• 1 Basics of Metric Spaces
  • 1.1 Metric Spaces
    • 1.1.1 Metric spaces: definition and examples
    • 1.1.2 Open and closed sets
    • 1.1.3 Convergence and continuity
  • 1.2 Useful properties of metric spaces
    • 1.2.1 Cauchy sequences and completeness
    • 1.2.2 Compactness
• 2 Basics of Linear Spaces
  • 2.1 Banach spaces (basic definitions only)
  • 2.2 Hilbert spaces
  • 2.3 Subspaces
  • 2.4 Linear spans
• 3 Orthogonality
  • 3.1 Orthogonal System in Hilbert Space
  • 3.2 Bessel’s inequality
  • 3.3 The Riesz–Fischer theorem
  • 3.4 Construction of Orthonormal Sequences
  • 3.5 Orthogonal complements
• 4 Duality of Linear Spaces
  • 4.1 Dual space of a normed space
  • 4.2 Self-duality of Hilbert space
• 5 Fourier Analysis
  • 5.1 Fourier series
  • 5.2 Fejér’s theorem
  • 5.3 Parseval’s formula
  • 5.4 Some Application of Fourier Series
• 6 Operators
  • 6.1 Linear operators
  • 6.2 Orthoprojections
  • 6.3 B(H) as a Banach space (and even algebra)
  • 6.4 Adjoints
  • 6.5 Hermitian, unitary and normal operators
• 7 Spectral Theory
  • 7.1 The spectrum of an operator on a Hilbert space
  • 7.2 The spectral radius formula
  • 7.3 Spectrum of Special Operators
• 8 Compactness
  • 8.1 Compact operators
  • 8.2 Hilbert–Schmidt operators
• 9 Compact normal operators
  • 9.1 Spectrum of normal operators
  • 9.2 Compact normal operators
• 10 Integral equations
• 11 Banach and Normed Spaces
  • 11.1 Normed spaces
  • 11.2 Bounded linear operators
  • 11.3 Dual Spaces
  • 11.4 Hahn–Banach Theorem
  • 11.5 C(X) Spaces
• 12 Measure Theory
  • 12.1 Basic Measure Theory
  • 12.2 Extension of Measures
  • 12.3 Complex-Valued Measures and Charges
  • 12.4 Constructing Measures, Products
• 13 Integration
  • 13.1 Measurable functions
  • 13.2 Lebesgue Integral
  • 13.3 Properties of the Lebesgue Integral
  • 13.4 Integration on Product Measures
  • 13.5 Absolute Continuity of Measures
• 14 Functional Spaces
  • 14.1 Integrable Functions
  • 14.2 Dense Subspaces in L_p
  • 14.3 Continuous functions
  • 14.4 Riesz Representation Theorem
• 15 Fourier Transform
  • 15.1 Convolutions on Commutative Groups
  • 15.2 Characters of Commutative Groups
  • 15.3 Fourier Transform on Commutative Groups
  • 15.4 The Schwartz space of smooth rapidly decreasing functions
  • 15.5 Fourier Integral
• 16 Advances of Metric Spaces
  • 16.1 The Stone–Weierstrass Theorem
  • 16.2 Contraction mappings and fixed point theorems
    • 16.2.1 The Banach fixed point theorem
    • 16.2.2 Applications of fixed point theory: The Picard-Lindelöf Theorem
    • 16.2.3 Applications of fixed point theory: Inverse and Implicit Function Theorems
  • 16.3 The Baire Category Theorem and Applications
    • 16.3.1 The Baire’s Categories
    • 16.3.2 Banach–Steinhaus Uniform Boundedness Principle
    • 16.3.3 The open mapping theorem
    • 16.3.4 The closed graph theorem
  • 16.4 Semi-norms and locally convex topological vector spaces
• A Tutorial Problems
  • A.1 Tutorial problems I
  • A.2 Tutorial problems II
  • A.3 Tutorial Problems III
  • A.4 Tutorial Problems IV
  • A.5 Tutorial Problems V
  • A.6 Tutorial Problems VI
  • A.7 Tutorial Problems VII
• B Solutions of Tutorial Problems
  • B.1 Solution of Tuitorial Problem I
  • B.2 Solutions of Tutorial Problems II
  • B.3 Solutions of Tutorial Problems III
  • B.4 Solutions of Tutorial Problems IV
  • B.5 Solutions of Tutorial Problems V
  • B.6 Solutions of Tutorial Problems VI
  • B.7 Solutions of Tutorial Problems VII
• C Course in the Nutshell
  • C.1 Some useful results and formulae (1)
  • C.2 Some useful results and formulae (2)
• D Supplementary Sections
  • D.1 Reminder from Complex Analysis

Notations and Assumptions

ℤ₊, ℝ₊ denotes non-negative integers and reals.
x,y,z,… denotes vectors.
λ,µ,ν,… denotes scalars.
ℜ z, ℑ z stand for real and imaginary parts of a complex number z.

Integrability conditions

In this course, the functions we consider will be real or complex valued functions defined on the real line which are locally Riemann integrable. This means that they are Riemann integrable on any finite closed interval [a,b]. (A complex valued function is Riemann integrable iff its real and imaginary parts are Riemann-integrable.) In practice, we shall be dealing mainly with bounded functions that have only a finite number of points of discontinuity in any finite interval. We can relax the boundedness condition to allow improper Riemann integrals, but we then require the integral of the absolute value of the function to converge.

We mention this right at the start to get it out of the way. There are many fascinating subtleties connected with Fourier analysis, but those connected with technical aspects of integration theory are beyond the scope of the course. It turns out that one needs a “better” integral than the Riemann integral: the Lebesgue integral, and I commend the module, Linear Analysis 1, which includes an introduction to that topic which is available to MM students (or you could look it up in Real and Complex Analysis by Walter Rudin). Once one has the Lebesgue integral, one can start thinking about the different classes of functions to which Fourier analysis applies: the modern theory (not available to Fourier himself) can even go beyond functions and deal with generalized functions (distributions) such as the Dirac delta function which may be familiar to some of you from quantum theory.

From now on, when we say “function”, we shall assume the conditions of the first paragraph, unless anything is stated to the contrary.

0 Motivating Example: Fourier Series

0.1  Fourier series: basic notions

Before proceed with an abstract theory we consider a motivating example: Fourier series.

0.1.1 2π-periodic functions

In this part of the course we deal with functions (as above) that are periodic.

We say a function f:ℝ→ℂ is periodic with period T>0 if f(x+T)= f(x) for all x∈ ℝ. For example, sinx, cosx, e^ix(=cos x+i sinx) are periodic with period 2π. For k∈ R∖{0}, sinkx, coskx, and e^ikx are periodic with period 2π/|k|. Constant functions are periodic with period T, for any T>0. We shall specialize to periodic functions with period 2π: we call them 2π-periodic functions, for short. Note that cosnx, sinnx and e^inx are 2π-periodic for n∈ℤ. (Of course these are also 2π/|n|-periodic.)

Any half-open interval of length T is a fundamental domain of a periodic function f of period T. Once you know the values of f on the fundamental domain, you know them everywhere, because any point x in ℝ can be written uniquely as x=w+nT where n∈ ℤ and w is in the fundamental domain. Thus f(x) = f(w+(n−1)T +T)=⋯ =f(w+T) =f(w).

For 2π-periodic functions, we shall usually take the fundamental domain to be ]−π, π]. By abuse of language, we shall sometimes refer to [−π, π] as the fundamental domain. We then have to be aware that f(π)=f(−π).

0.1.2 Integrating the complex exponential function

We shall need to calculate ∫_a^b e^ikx d x, for k∈ℝ. Note first that when k=0, the integrand is the constant function 1, so the result is b−a. For non-zero k, ∫_a^b e^ikx d x= ∫_a^b (coskx+isinkx) d x = (1/k)[ (sinkx − icoskx)]_a^b = (1/ik)[(coskx+isinkx)]_a^b = (1/ik)[e^ikx]_a^b = (1/ik)(e^ikb−e^ika). Note that this is exactly the result you would have got by treating i as a real constant and using the usual formula for integrating e^ax. Note also that the cases k=0 and k≠0 have to be treated separately: this is typical.

This follows immediately from Ex. 2 and Prop.4.

For real-valued functions, the introduction of complex exponentials seems artificial: indeed they can be avoided as follows. We work with (1) in the case of a finite sum: then we can rearrange the sum as

Here

for n>0 and

for n>0. a₀ = 1/π∫_−π^πf(x) d x, the constant chosen for consistency.

The a_n and b_n are also called Fourier coefficients: if it is necessary to distinguish them, we may call them Fourier cosine and sine coefficients, respectively.

We note that if f is real-valued, then the a_n and b_n are real numbers and so ℜ f(n) = ℜ f(−n), ℑ f(−n) = −ℑf(n): thus f(−n) is the complex conjugate of f(n). Further, if f is an even function then all the sine coefficients are 0 and if f is an odd function, all the cosine coefficients are zero. We note further that the sine and cosine coefficients of the functions coskx and sinkx themselves have a particularly simple form: a_k=1 in the first case and b_k=1 in the second. All the rest are zero.

For example, we should expect the 2π-periodic function whose value on ]−π,π] is x to have just sine coefficients: indeed this is the case: a_n=0 and b_n=i(f(n)−f(−n)) = (−1)ⁿ⁺¹2/n for n>0.

The above question can then be reformulated as “to what extent is f(x) represented by the Fourier series a₀/2 + ∑_n>0(a_ncosx + b_nsinx)?” For instance how well does ∑(−1)ⁿ⁺¹(2/n)sinnx represent the 2π-periodic sawtooth function f whose value on ]−π, π] is given by f(x) = x. The easy points are x=0, x=π, where the terms are identically zero. This gives the ‘wrong’ value for x=π, but, if we look at the periodic function near π, we see that it jumps from π to −π, so perhaps the mean of those values isn’t a bad value for the series to converge to. We could conclude that we had defined the function incorrectly to begin with and that its value at the points (2n+1)π should have been zero anyway. In fact one can show (ref. ) that the Fourier series converges at all other points to the given values of f, but I shan’t include the proof in this course. The convergence is not at all uniform (it can’t be, because the partial sums are continuous functions, but the limit is discontinuous.) In particular we get the expansion

which can also be deduced from the Taylor series for tan⁻¹.

0.2 The vibrating string

In this subsection we shall discuss the formal solutions of the wave equation in a special case which Fourier dealt with in his work.

We discuss the wave equation

subject to the boundary conditions

for all t≥0, and the initial conditions

This is a mathematical model of a string on a musical instrument (guitar, harp, violin) which is of length π and is plucked, i.e. held in the shape F(x) and released at time t=0. The constant K depends on the length, density and tension of the string. We shall derive the formal solution (that is, a solution which assumes existence and ignores questions of convergence or of domain of definition).

0.2.1 Separation of variables

We first look (as Fourier and others before him did) for solutions of the form y(x,t) = f(x)g(t). Feeding this into the wave equation (2) we get

and so, dividing by f(x)g(t), we have

The left-hand side is an expression in x alone, the right-hand side in t alone. The conclusion must be that they are both identically equal to the same constant C, say.

We have f^′′(x) −Cf(x) =0 subject to the condition f(0) = f(π) =0. Working through the method of solving linear second order differential equations tells you that the only solutions occur when C = −n² for some positive integer n and the corresponding solutions, up to constant multiples, are f(x) = sinnx.

Returning to equation (4) gives the equation g^′′(t)+K²n²g(t) =0 which has the general solution g(t) = a_ncosKnt + b_nsinKnt. Thus the solution we get through separation of variables, using the boundary conditions but ignoring the initial conditions, are

for n≥ 1.

0.2.2 Principle of Superposition

To get the general solution we just add together all the solutions we have got so far, thus

ignoring questions of convergence. (We can do this for a finite sum without difficulty because we are dealing with a linear differential equation: the iffy bit is to extend to an infinite sum.)

We now apply the initial condition y(x,0) = F(x) (note F has F(0) =F(π) =0). This gives

We apply the reflection trick: the right-hand side is a series of odd functions so if we extend F to a function G by reflection in the origin, giving

we have

for −π≤ x ≤ π.

If we multiply through by sinrx and integrate term by term, we get

so, assuming that this operation is valid, we find that the a_n are precisely the sine coefficients of G. (Those of you who took Real Analysis 2 last year may remember that a sufficient condition for integrating term-by -term is that the series which is integrated is itself uniformly convergent.)

If we now assume, further, that the right-hand side of (5) is differentiable (term by term) we differentiate with respect to t, and set t=0, to get

This equation is solved by the choice b_n=0 for all n, so we have the following result

0.3 Historic: Joseph Fourier

Joseph Fourier, Civil Servant, Egyptologist, and mathematician, was born in 1768 in Auxerre, France, son of a tailor. Debarred by birth from a career in the artillery, he was preparing to become a Benedictine monk (in order to be a teacher) when the French Revolution violently altered the course of history and Fourier’s life. He became president of the local revolutionary committee, was arrested during the Terror, but released at the fall of Robespierre.

Fourier then became a pupil at the Ecole Normale (the teachers’ academy) in Paris, studying under such great French mathematicians as Laplace and Lagrange. He became a teacher at the Ecole Polytechnique (the military academy).

He was ordered to serve as a scientist under Napoleon in Egypt. In 1801, Fourier returned to France to become Prefect of the Grenoble region. Among his most notable achievements in that office were the draining of some 20 thousand acres of swamps and the building of a new road across the alps.

During that time he wrote an important survey of Egyptian history (“a masterpiece and a turning point in the subject”).

In 1804 Fourier started the study of the theory of heat conduction, in the course of which he systematically used the sine-and-cosine series which are named after him. At the end of 1807, he submitted a memoir on this work to the Academy of Science. The memoir proved controversial both in terms of his use of Fourier series and of his derivation of the heat equation and was not accepted at that stage. He was able to resubmit a revised version in 1811: this had several important new features, including the introduction of the Fourier transform. With this version of his memoir, he won the Academy’s prize in mathematics. In 1817, Fourier was finally elected to the Academy of Sciences and in 1822 his 1811 memoir was published as “Théorie de la Chaleur”.

For more details see Fourier Analysis by T.W. Körner, 475-480 and for even more, see the biography by J. Herivel Joseph Fourier: the man and the physicist.

What is Fourier analysis. The idea is to analyse functions (into sine and cosines or, equivalently, complex exponentials) to find the underlying frequencies, their strengths (and phases) and, where possible, to see if they can be recombined (synthesis) into the original function. The answers will depend on the original properties of the functions, which often come from physics (heat, electronic or sound waves). This course will give basically a mathematical treatment and so will be interested in mathematical classes of functions (continuity, differentiability properties).

1 Basics of Metric Spaces

1.1 Metric Spaces

1.1.1 Metric spaces: definition and examples

In Analysis and Calculus the definition of convergence was based on the notion of a distance between points, namely the standard distance between two real numbers is given by

Similarly, the distance between two points in the plane, given by

A metric space formalises this notion. This will give us the flexibility to talk about distances on function spaces, for example, or introduce other notions of distance on spaces.

There is a natural name for a class of maps, which preserve metrics:

1.1.2 Open and closed sets

Trivial but useful observations are:

• x ∈ B_r(x) ⊂ B_r(x) for all x∈ X and r>0, so neither ball is empty and every point is covered by all balls it centres;
• B_r(x) ⊂ B_r+ε (x) for all x∈ X, r>0 and whatever small ε>0.

Note, that in ℝ with the usual metric the open ball is B_r(x)=(x−r,x+r), an open interval, and the closed ball is B_r(x)=[x−r,x+r], a closed interval.

For the d₂ metric on ℝ², the unit ball, B₁(0), is disc centred at the origin, excluding the boundary. You may like to think about what you get for other metrics on ℝ². What are balls in the discrete metric, Ex. 2.2?

Clearly X itself is an open set, that is the whole metric space is open in itself. Also the empty set ∅ is also considered to be open in a trivial way.

Note that closed does not mean “not open". In a metric space the sets ∅ and X are both open and closed. In ℝ we have:

• (a,b) is open.
• [a,b] is closed, since its complement (−∞,a) ∪ (b,∞) is open.
• [a,b) is not open, since there is no open ball B(a,r) contained in the set. Nor it is closed, since its complement (−∞,a) ∪ [b,∞) isn’t open (no ball centred at b can be contained in the set).

Thereafter, the collection of open sets is preserved by arbitrary unions and finite intersections.

However, an arbitrary intersection of open sets is not always open; for example (−1/n,1/n) is open for each n=1,2,3,…, but ∩_n=1^∞(−1/n,1/n)= {0}, which is not an open set.

For closed sets we swap union and intersection.

Infinite unions of closed sets do not need to be closed. An example is

which is open but not closed in ℝ with standard metric.

The above smallest closed set containing S does exist, because we can define

the intersection of all closed sets containing S. There is at least one closed set containing S, namely X itself.

We see that S is open exactly when S=intS, otherwise intS is smaller.

1.1.3 Convergence and continuity

Let (x_n) be a sequence in a metric space (X,d), i.e., x₁,x₂,…. (Sometimes we may start counting at x₀.)

This is the usual notion of convergence if we think of points in ℝ^d with the Euclidean metric.

This can also be phrased as follows, characterising closed set in terms of sequences.

Hence, the closure S is obtained from S by adding all possible limit points of sequences in S.

A result on convergence in ℝ^m.

A similar result holds for ℝ^m in general.

Now let’s look at continuous functions again.

Now we look at continuous functions between general metric spaces.

Another way of saying the same is that for every ε>0 there exists a δ>0 such that

The map f is continuous, if it is continuous at all points of X.

But there is a nicer way to define continuity. For a mapping f: X → Y and a set U ⊂ Y, let f⁻¹(U) be the set, called pre-image or inverse image

This makes sense even if f⁻¹ is not defined as a function.

In many cases we may need a stronger notion.

Note, that here the same δ_ε shall work for all x∈ X. Thus any uniformly continuous function is continuous at every point. On the other hand the function f(x)=1/x on (0,1) is continuous but not uniformly continuous.

1.2 Useful properties of metric spaces

Metric spaces may or may not have some useful properties which we are discussing in the following subsections: completeness and compactness.

1.2.1 Cauchy sequences and completeness

Recall that if (X,d) is a metric space, then a sequence (x_n) of elements of X converges to x∈ X if d(x_n,x) → 0, i.e., if given ε>0 there exists N such that d(x_n,x)< ε whenever n ≥ N. Thus, to show that a sequence is convergent from the definition we need to present its limit x which may not belong to the sequence (x_n). It would be convenient to deduce convergence of (x_n) just through its own properties without a reference to extraneous x. This is possible for complete metric spaces studied in this subsection.

Often we think of convergent sequences as ones where x_n and x_m are close together when n and m are large. This is almost, but not quite, the same thing in a general metric space.

In fact all convergent sequences are Cauchy sequences, by the following result.

Does every Cauchy sequence converge?

This is a result from the first year. Since its proof depends on the definition of ℝ we will not demonstrate it here.

If a metric space (X,d) is not complete one can always pass to its abstract completion in the following sense.

There are many important consequences of Ex. 62, in particular the following.

1.2.2 Compactness

Accordingly to a dictionary: compact—closely and firmly united or packed together. For a metric space a meaning of “closely and firmly united” can be defined in several different forms—through open coverings or convergent subsequences—and we will see that these interpretations are equivalent.

An open cover of a metric space (X,d) is a family of open sets (U_α)_{α ∈ I} such that

A subcover of a cover is a subset I′ ⊂ I of the index set such that (U_α)_{α ∈ I′} is still a cover.

Informally: a space is compact if any infinite open covering is excessive and can be reduced just to a finite one. An example of a compact set is [0,1] and example of non-compact—all reals or the open interval (0,1). An importance of this concept is clarified by Rem. 21.

The limit of a convergent sequence is called the accumulation point of {x_n}. It is instructive to compare the definitions of:

Thereafter, x is not an accumulation point of {x_n} if for some ε>0 and some N for all subsequent n>N we have d(x, x_n)>ε.

Informally: a space is sequentially compact if there is no room to place infinite number of points sufficiently apart from each other to avoid their condensation to a limit. Taking the sequence x_n=n shows that the set of all reals is not sequentially compact. On the other hand, we know from previous years that bounded closed set in ℝⁿ every sequence has a convergent subsequence. Therefore, bounded closed sets in ℝⁿ are sequentially compact.

The converse of this statement is not correct in general. It is however famously correct in ℝ^m.

As an illustration of further nice properties of compact spaces we mention the following result:

2 Basics of Linear Spaces

  A person is solely the concentration of an infinite set of interrelations with another and others, and to separate a person from these relations means to take away any real meaning of the life.

 Vl. Soloviev

A space around us could be described as a three dimensional Euclidean space. To single out a point of that space we need a fixed frame of references and three real numbers, which are coordinates of the point. Similarly to describe a pair of points from our space we could use six coordinates; for three points—nine, end so on. This makes it reasonable to consider Euclidean (linear) spaces of an arbitrary finite dimension, which are studied in the courses of linear algebra.

The basic properties of Euclidean spaces are determined by its linear and metric structures. The linear space (or vector space) structure allows to add and subtract vectors associated to points as well as to multiply vectors by real or complex numbers (scalars).

The metric space structure assign a distance—non-negative real number—to a pair of points or, equivalently, defines a length of a vector defined by that pair. A metric (or, more generally a topology) is essential for definition of the core analytical notions like limit or continuity. The importance of linear and metric (topological) structure in analysis sometime encoded in the formula:

On the other hand we could observe that many sets admit a sort of linear and metric structures which are linked each other. Just few among many other examples are:

• The set of convergent sequences;
• The set of continuous functions on [0,1].

It is a very mathematical way of thinking to declare such sets to be spaces and call their elements points.

But shall we lose all information on a particular element (e.g. a sequence {1/n}) if we represent it by a shapeless and size-less “point” without any inner configuration? Surprisingly not: all properties of an element could be now retrieved not from its inner configuration but from interactions with other elements through linear and metric structures. Such a “sociological” approach to all kind of mathematical objects was codified in the abstract category theory.

Another surprise is that starting from our three dimensional Euclidean space and walking far away by a road of abstraction to infinite dimensional Hilbert spaces we are arriving just to yet another picture of the surrounding space—that time on the language of quantum mechanics.

  The distance from Manchester to Liverpool is 35 miles—just about the mileage in the opposite direction!

  A tourist guide to England

2.1 Banach spaces (basic definitions only)

The following definition generalises the notion of distance known from the everyday life.

The following notion is a useful specialisation of metric adopted to the linear structure.

The connection between norm and metric is as follows:

---

  (a)    (b)    

Figure 1: Triangle inequality in metric (a) and normed (b) spaces.

---

An important notions known from real analysis are limit and convergence. Particularly we usually wish to have enough limiting points for all “reasonable” sequences.

For example, the set of integers ℤ and reals ℝ with the natural distance functions are complete spaces, but the set of rationals ℚ is not. The complete normed spaces deserve a special name.

---

(i)    (ii)   (iii)

Figure 2: Different unit balls defining norms in ℝ2 from Example 9.

---

  —We need an extra space to accommodate this product!

  A manager to a shop assistant

2.2 Hilbert spaces

Although metric and norm capture important geometric information about linear spaces they are not sensitive enough to represent such geometric characterisation as angles (particularly orthogonality). To this end we need a further refinements.

From courses of linear algebra known that the scalar product ⟨ x,y ⟩= x₁ y₁ + ⋯ + x_n y_n is important in a space ℝⁿ and defines a norm ||x||²=⟨ x,x ⟩. Here is a suitable generalisation:

Last two properties of the scalar product is oftenly encoded in the phrase: “it is linear in the first variable if we fix the second and anti-linear in the second if we fix the first”.

Now we state, probably, the most important inequality in analysis.