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- Preliminaries This preliminary chapter is an attempt to bring together important results from earlier courses which we shall need this term. It is unlikely that I shall manage to include everything at the first attempt. So the chapter may grow during the course of the term as I think of more things that ought to have been included.
0.1. Topology. I shall assume students attending this course know the basics of topology, as to be found in the second- year Section A course. What follows are a few reminders, intended mainly to establish some notation.
Definition 0.1. A topology on a set X is a subset T of the power set PX [that is to say, a collection T of subsets of X] satisfying:
TOP1 ∅ ∈ T , X ∈ T ; TOP2 S ⊆ T =⇒
S ∈ T ;
TOP3 U, V ∈ T =⇒ U ∩ V ∈ T.
Of course, it follows from TOP3 that if A is a finite subset of T then
A ∈ T. We may thus summarize TOP2 and TOP3 as stating that T is closed under arbitrary unions and finite intersections.
I am in too much of a hurry to write out the (I hope familiar) definitions of continuity, compactness, closure and so on. As a matter of notation, I may use either of the notations A or cl A for the closure of a subset A in a topological space. Recall that a subset A of X is said to be dense if its closure is the whole of X.
Definition 0.2. A metric on a set X is a function d : X × X → [0, ∞) satisfying:
D1 d(x, y) = 0 =⇒ x = y; D2 d(x, y) = d(y, x) for all x, y ∈ X; D3 d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
We define the open ball B(x; r) (writing BdX (x; r) if there may be ambiguity about what the metric is and what the “ambient space” X is) and the closed ball B[x; r] with centre x and radius r by
B(x; r) = {y ∈ X : d(x, y) < r} B[x; r] = {y ∈ X : d(x, y) ≤ r}
The open balls form a base for the topology Td. A topology T is said to be metrizable if there is some metric d with T = Td.
Metrizable topologies have some nice properties, notably those allowing a description of topological properties like continuity, closure and compactness in terms of convergent sequences; I hope you are familiar with these. Be warned, however, that not all of the topologies that occur in functional analysis are metrizable. (See, for instance, the weak and weak* topologies of Chapter 5) Here is one important feature of metrizable topologies that may have slipped your attention. Recall that a topological space X is said to be separable if there is a countable subset X which is dense in X.
Proposition 0.3. If X is a separable metrizable space and Y is a subset of X then Y is separable (for the subspace topology).
Look up a proof if you need to. Just note that the “obvious approach”, which starts let D be countable and dense in X, and consider D ∩ Y , does not work (D ∩ Y is frequently empty). For general topological spaces a subset of a separable space need not be separable. 1
0.2. Integration. It is not my intention to give a course on Measure and Integration, but we shall have to be ready to use results from that subject in a rather more formal way than you will have experienced in the Section A course.
Definition 0.4. Let Ω be a non-empty set. We say that a collection of subsets F of Ω is a σ-field (or σ-algebra) if the following hold:
σF1 ∅ ∈ F ; σF2 F ∈ F =⇒ Ω \ F ∈ F ; σF3 Fn ∈ F (n ∈ N) =⇒
n∈N Fn^ ∈^ F^. Thus a σ-field F is closed under complementation and countable unions. It follows from de Morgan’s laws that F is also closed under countable intersections. For any collection A of subsets of a set Ω there is a smallest σ-field σA containing A. In particular, when (L, T ) is a topological space, there is a smallest σ-field containing T ; this is called the Borel σ-field of L.
Definition 0.5. Let F be a σ-field of subsets of a set Ω. A function μ : F → [0, ∞] is said to be a measure on F if:
M1 μ(∅) = 0; (^1) Indeed the study of hereditarity separable spaces is “a can of worms”.
2
M2 μ
n∈N Fn
n∈N μ(Fn) whenever^ F^1 , F^2 ,...^ are^ pairwise disjoint^ members of^ F^. We then say that (Ω, F , μ) is a measure space. If μ(Ω) = 1 we say that μ is a probability measure and that (Ω, F , μ) is a probability space. In this case the notation P is usually used instead of μ.
Examples 0.6.
(1) On any set Ω we may take F to be the collection P(Ω) of all subsets of Ω define the “counting measure” # by setting #(F ) = n if F is a finite set with n elements and #(F ) = ∞ if F is infinite. (2) On Ω = R we may take F to be the collection MLeb of all Lebesgue measurable sets, and μ to be Lebesgue measure. (3) Taking Ω to be the unit interval (0, 1), and P to be the restriction of Lebesgue to the subsets of (0, 1) we have an important example of a probability space. (4) Let Ω be any uncountable set, take F = P(Ω) and define
μ(F ) =
0 if F is countable ∞ if F is uncountable.
This is a nasty example and we usually make assumptions on our measure spaces to exclude such pathologies. The commonest such assumption is given below as Definition 0.7.
Definition 0.7. We say that a measure space (Ω, F , μ) is σ-finite if there exist An ∈ F with Ω =
n∈N An^ and μ(An) < ∞ for all n.
From now on, we shall assume that all our measure spaces are σ-finite.
Definition 0.8. A real-valued function x on a measure space (Ω, F , μ) is said to be measurable (F -measurable if we are being careful) if {t ∈ Ω : x(t) ≤ α} = x−^1 (−∞, α] ∈ F for every real α. It follows that x−^1 [B] ∈ F for every Borel subset of R. Borrowing notation from probability theory, we shall write
[x ≤ α] = x−^1 (−∞, α], [x ∈ B] = x−^1 [B],
and so on. For any F ∈ F we define the indicator function (^1) F by setting
(^1) F (t) =
1 if t ∈ F 0 if t /∈ F.
A function g : Ω → R is called an F - simple function if we can write
g =
∑^ n
j=
αj 1 Fj ,
where α 1 ,... , αn are real numbers and F 1 ,... , Fn are members of F with μ(Fj ) < ∞.
Definition 0.9. For an F -simple function g =
∑n j=1 αj^1 Fj^ we (well!) define ∫
Ω
g dμ =
∑^ n
j=
αj μ(Fj ).
For a non-negative real-valued F -measurable function x on Ω, we define ∫
Ω
x dμ = sup{
Ω
g dμ : g ≤ x and g is F -simple} ∈ [0, ∞]
We say that x is integrable if this supremum is finite. For a general real-valued F -measurable function x, we say that x is integrable, if both x+^ and x−^ are, and we set ∫
Ω
x dμ =
Ω
x+^ dμ −
Ω
x−^ dμ.
When (Ω, F , P) is a probability space, it is usual to refer to refer to measurable functions as random variables and to use the notation E[x] for
x dP; we call E[x] the expectation of x. In all cases, the set L 1 (Ω, F , μ) of μ-integrable functions is a vector space and the integral is a linear mapping. There are two important convergence theorems.
Theorem 0.10 (The Monotone Convergence Theorem). Let (xn)n∈N be an increasing sequence of μ-integrable functions with supn
xn dμ < ∞. Then the sequence (xn(t))n∈N converges (in R) for almost all values of t and the function x defined (almost everywhere) by x(t) = limn xn(t) is integrable with
x dμ = lim
xn dμ.
(3) For 1 ≤ p < ∞ the space `p is defined to consist of all p-summable sequences (i.e. sequences (an)n∈N such that the series
n=1 |an|
p (^) converges. For such a sequence we define
‖(a 1 , a 2 , a 3 ,... )‖p =
j=
|aj |p
) 1 /p .
(4) If L is a topological space we define C b(L) to be the space of all bounded continuous scalar-valued functions on L, and equip it with the supremum norm ‖x‖∞ = sup t∈L
|x(t)|.
When L is compact, all functions on L are bounded so we may leave out the superscript b and write simply C (L). The norm axioms are easy to verify for all of the cases in 0.14 except for the `p-norms where p /∈ { 1 , 2 , ∞}. For these, we need Minkowski’s Inequality (∑
j
|aj + bj |p
) 1 /p ≤
j
|aj |p
) 1 /p
j
|bj |p
) 1 /p .
We shall give a proof of this in Chapter 2. Another important class of Banach spaces are the Lebesgue spaces Lp. These may already be familiar to you, but we shall give a detailed treatment in a later section. Given a norm on a space X, we can define a metric by d(x, y) = ‖x − y‖ and introduce the associated topology, thus giving meaning to such notions as continuity of mappings, and completeness, separability, compactness, precompactness for X and its subsets. We retain the ball notation, putting BX (x; r) = {y ∈ X : ‖x − y‖ < r} and BX [x; r] = {y ∈ X : ‖x − y‖ ≤ r}. It has become standard to write BX for BX [0; 1] and to refer to this as “the” unit ball of X. The unit sphere {x ∈ X : ‖x‖ = 1} is denoted SX. Given two norms ‖ · ‖ and ‖ · · · ‖′^ on the same space X, we say that ‖ · ‖ dominates ‖ · ‖′^ if there exists a positive constant C such that ‖x‖′^ ≤ C‖x‖ for all x. If in addition there exists C′^ such that ‖x‖ ≤ C′‖x‖′, we say that the two norms are equivalent. On the space Rn^ we may (fairly) easily see that
‖(a 1 , a 2 ,... , an)‖q ≤ ‖(a 1 , a 2 ,... , an)‖p ≤ C‖(a 1 , a 2 ,... , an)‖q ,
where C = n^1 −p/q^ when p ≤ q. In particular, all the `p norms are equivalent on Rn^ (though with constants that get worse as n increases. In fact, more is true.
Proposition 0.15. On a finite dimensional space X all norms are equivalent.
Corollary 0.16. If X is finite-dimensional then closed bounded subsets of X are compact.
It is very important to remember that closed bounded sets are not compact in general. Indeed we have the following converse to the above corollary.
Proposition 0.17. If X is a normed space and the unit ball BX is precompact then X is finite-dimensional.
Exercise 0.18. Let c 00 denote the vector space of sequences a = (a 1 , a 2 , a 3 ,... ) such that all but finitely many of the terms aj are zero. Then c 00 may be equipped with any of the `p-norms of 0.14(2) and (3). We see (fairly) easily that ‖ · ‖p dominates ‖ · ‖q if and only if p ≤ q. These norms are thus non-equivalent when p 6 = q.
Definition 0.19. A normed space which is complete is called a Banach space. All of the spaces in 0.14 are Banach spaces. By Proposition 0.15 every finite-dimensional normed space is complete. A complete inner-product space is called a Hilbert space; ` 2 ad L 2 are Hilbert spaces.
Definition 0.20. Let xn (n ∈ N) be elements of a normed space X. We say that the series
n=1 converges^ in^ X^ if there exists s ∈ X such that ‖s −
∑n j=1 xj^ ‖ →^ 0 as^ n^ → ∞. We say that the series^
n=1 xn^ converges absolutely^ if the scalar series
n=1 ‖xn‖^ is convergent. It is a familiar fact from Mods Analysis that absolute convergence implies convergence for scalar series. Here is the normed space version of that result.
Proposition 0.21. Let X be a normed space. Then X is a Banach space if and only if every absolutely convergent series in X is convergent in X.
Proposition 0.22. Let X and Y be normed spaces and let T : X → Y be a linear mapping. Then the following are equivalent:
(1) T is continuous; (2) T is continuous at 0 ;
(3) there exists a constant M such that ‖T (x)‖ ≤ M ‖x‖ for all x ∈ X.
Definition 0.23. A linear mapping T satisfying the equivalent conditions of Proposition 0.22 will be called a bounded linear operator (often abbreviated to BLO). The smallest constant M that works in 0.22(3) will be called the operator norm of T and will be denoted ‖T ‖. The space of all bounded linear operators from X to Y will be denoted L(X; Y ); L(X; X) is usually abbreviated to L(X); L(X; K) is denoted X∗^ and is called the dual space of X. The elements of X∗ are usually referred to as bounded (or continuous) linear “functionals” on X.
The reader should be warned that there are other notations in popular use: some authors prefer B(X) rather than L(X) (emphasizing “boundedness”, rather than linearity); others would use X′^ for our X∗.
Proposition 0.24. When X and Y are normed spaces, L(X; Y ) is a normed space when equipped with the operator norm. If Y is a Banach space so is L(X; Y ). In particular X∗^ is always a Banach space.
Examples 0.25.
(1) The dual of c 0 is, in a natural way, isometrically isomorphic to 1. (2) For 1 ≤ p < ∞ the dual ofp is, in a natural way, isometrically isomorphic to `q with 1/p + 1/q = 1. (3) A similar result holds for Lp and we shall prove it in Chapter 2. (4) For any real Hilbert space there is a natural isometric isomorphism between H and H∗^ (something similar happens for complex Hilbert spaces but complex conjugation is involved); (5) There is a representation theorem for the dual space of C (K) when K is compact in terms of “signed measures” the Borel σ-field of K. We do not have room for this useful result in the present course.
Definition 0.26. Let X and Y be normed spaces and let T : X → Y be linear. We say that T is an isomorphism if both T and T −^1 are continuous. [Thus an isomorphism is a linear homeomorphism.] An equivalent condition is that T be surjective and that there exist positive constants A, B such that
A−^1 ‖x‖ ≤ ‖T (x)‖ ≤ B‖x‖
for all x ∈ X. If T satisfies such an inequality, but is not necessarily surjective, we say that T is an isomorphic embedding. We say that T is an isometric isomorphism (or embedding) if ‖T (x)‖ = ‖x‖ for all x.
A powerful technique in functional analysis is the use of dense linear subspaces. When A is a subset of a normed space X, we write sp〈A〉 for the linear subspace generated by A and sp〈A〉 for its closure (also a linear subspace). A normed space is separable if there exists a countable subset A with sp〈A〉 = X.
Examples 0.27.
(1) If X = p (1 ≤ p < ∞ or c 0 the unit vectors en = (0, 0 , 0 ,... , 0 , 1 , 0 ,... ), (with 1 in the nth^ coordinate) form a subset with dense linear span, i.e. sp〈en : n ∈ N〉 = X. Note that this is not the case if X =∞ or X = c. (2) If X = Lp(R) (1 ≤ p < ∞) then we may take A to consist of all indicator functions (^1) I with I a bounded interval. Then sp〈A〉 is the space of all step functions (more precisely their equivalence classes modulo null functions) and sp〈A〉 = Lp(R). If we restrict attention to intervals I with rational end-points we obtain a countable subset A with sp〈A〉 = Lp(R), thus establishing separability of Lp(R). (3) In the space C [0, 1] the polynomials form a dense linear subspace (though this has not yet been proved in a course you have attended). Another interesting dense linear subspace is formed by the piecewise-linear functions. In this case, density is easy to prove (using uniform continuity).
Theorem 0.28 (Uniqueness theorem). Let X and Y be normed spaces, let T 1 and T 2 be bounded linear operators from X to Y and let A be a subset of X with sp〈A〉 = X. If T 1 A= T 2 A then T 1 = T 2.
Proof. Given our hypotheses, ker(T 2 − T 1 ) is a closed linear subspace containing A and hence containing sp〈A〉 = X.
Theorem 0.29 (Continuous extension). Let X be a normed space, let Y be a Banach space, let Z be a dense linear subspace of X and let T : Z → Y be a bounded linear operator. Then there is a unique bounded linear operator U : X → Y extending T. We have ‖U ‖ = ‖T ‖. If T is an isomorphic (resp. isometric) embedding then so is U.
Theorem 0.30 (Convergence theorem). Let X be a normed space, let A be a subset of X with dense linear span, let Y be a Banach space and let (Tn)n∈N be a sequence of bounded linear operators from X to Y. Assume
(1) supn∈N ‖Tn‖ < ∞; (2) Tn(x) converges to a limit in Y for every x ∈ A.
Then there is a bounded linear operator T : X → Y such that ‖Tn(x) = T (x)‖ → 0 for all x ∈ X.
Theorem 0.31 (Hahn–Banach Extension Theorem). Let Z be a linear subspace of a normed space X and let g ∈ Z∗. Then there exists f ∈ X∗^ with ‖f ‖ = ‖g‖ and f Z = g.
Proof of Proposition 0.38. We shall prove [HI] and [MI], since the inequalities for sums may be regarded as special cases (by considering the counting measure). So let x ∈ Lp and y ∈ Lq and assume initially that ‖x‖p = ‖y‖q = 1. For each s ∈ Ω we apply the lemma to the positive real numbers |x(s)|, |y(s)| obtaining
|x(s)y(s)| ≤
|x(s)|p p
|y(s)|q q
Integrating, we obtain
|
Ω
x(s)y(s) dμ(s)| ≤
Ω
|x(s)y(s)| dμ(s) ≤
Ω
|x(s)|p p
|y(s)|q q
dμ(s) =
‖x‖pp p
‖y‖qq q
For general x and y we introduce the normalized versions ˆx = ‖x‖− p 1 x and ˆy = ‖y‖− q 1 y for which we have
|
x ˆˆy dμ| ≤ 1.
This immediately yields [HI]. We now pass to [MI]. Let x and y be in Lp(μ). We notice that
‖x + y‖pp =
|x(s) + y(s)|pdμ(s) =
(x(s) + y(s))z(s) dμ(s) =
x(s)z(s) dμ(s) +
y(s)z(s) dμ(s),
where z(s) = sign (x(s) + y(s))|x(s) + y(s)|p−^1. Now this function z is in Lq with
‖z‖qq =
|z|q^ dμ =
|x + y|q(p−1)^ =
|x + y|p^ = ‖x + y‖pp.
[This is part of the magic of conjugate indices.] Applying [HI] twice we obtain
‖x + y‖pp =
|x(s) + y(s)|pdμ(s) =
x(s)z(s) dμ(s) +
y(s)z(s) dμ(s) ≤ ‖x‖p‖z‖q + ‖y‖p‖z‖q ,
which, after dividing by ‖z‖q = ‖x + y‖ p/q p , becomes ‖x + y‖p = ‖x + y‖pp− p/q= ‖x + y‖pp‖z‖− q 1 ≤ ‖x‖p + ‖y‖p,
as required.
Exercise 0.40. Investigate the conditions under which equality occurs in the inequalities of H¨older and Minkowski.
Proposition 0.41. When 1 ≤ p < ∞, the space Lp(μ) is a Banach space under the norm ‖ · ‖p.
Proof. We have already remarked in Definition 0.37 that ‖x•‖p is well-defined. Let’s just check on this:
x•^ = y•^ =⇒ x(t) = y(t) μ − a.e. =⇒
|x|pdμ =
|y|pdμ.
The norm axiom N2 is obvious and N3 is Minkowski’s Inequality. We are left with N1. Suppose then that ‖x•‖p = 0. The non-negative integrable function |x|p^ thus satisfies ∫ |x|pdμ = 0.
So x(t) = 0 μ-a.e. by Proposition 0.12, x ∈ N and x•^ = 0. To prove completeness we shall use Proposition 0.21. Let x• n ∈ Lp(μ) be such that
n ‖xn‖p^ converges.^ Our first task is to show that the scalar series
n xn(t) converges a.e. We set^ sn^ =^
∑n j=1 xn(t),^ Sn^ =^
∑n j=1 |xn(t)|^ and note that Sn ∈ Lp(μ) with (∫ Snpdμ
) 1 /p ≤
∑^ n
j=
‖x• j ‖p
by Minkowski’s inequality. Setting un = Snp, we have an increasing sequence of μ-integrable functions satisfying ∫ un dμ ≤ M =
n=
‖x• n‖p
)p .
By the Monotone Convergence Theorem, un(t) tends to a finite limit u(t) almost everywhere, and the function u is integrable with
u dμ ≤ M. It follows that the scalar series
n |xn(t)|^ converges almost everywhere to a finite limit S(t) = u(t)^1 /p^ and that S ∈ Lp(μ) with
Spdμ ≤ M. Thus the series
n xn(t) converges a.e. (since absolute convergence implies convergence for scalar series) and the limit function s satisfies s ≤ S a.e.
So s•^ ∈ Lp with
‖s•‖p ≤ ‖S•‖p ≤
∑^ ∞
n=
‖x• n‖p.
To finish the completeness proof we need to check that ‖s − sm‖p → 0 as m → ∞. This is easy since the above argument shows that
‖s•^ − s• n‖p ≤
∑^ ∞
n=m+
‖x• n‖p.
For p = ∞ things are a little different, since the space L∞ of bounded measurable functions is already a Banach space for the supremum norm. (It’s a closed subspace of the Banach space `∞(Ω) of all bounded functions on Ω. However, we still have to take a quotient if we want to identify functions that differ only on a null set. This time the quotient is just a special case of Definition 1.4.
Definition 0.42. When (Ω, F , μ) is a measure space we define L∞(Ω, F , μ) to be the space of all bounded μ-measurable functions, equipped with the supremum norm. Let N∞ be the closed subspace consisting of those bounded functions that are almost everywhere zero [so N∞ = N ∩ L∞]. The quotient space L∞/N∞ is denoted L∞(μ) and is a Banach space equipped with the quotient norm.
Proposition 0.43. For x ∈ L∞ the quotient norm is given by
‖x•‖∞ = inf{λ ∈ R : |x(t)| ≤ λ a.e.}.
By abuse of notation, we shall generally leave out the “blob” when referring to a coset x•^ ∈ Lp. This convention requires a little care (particularly when p = ∞) but to do otherwise would leave us open to accusations of pedantry. When 0 < p < 1 the function ‖ · ‖p is not a norm (the triangle inequality fails). We should not assume from this that such values of p are without interest to analysts and probabilists. Losing the triangle inequality is not the end of the world.
0.5. Probability. Probability theory is far from being “just” a special case of measure theory. As well as its important applications in the “real world”, probability has a lot to offer analysts, including those of us who are interested in Banach spaces and operators. Thinking about elements of L 1 [0, 1] as random variables allows us to introduce notions like independence, distribution functions and characteristic functions and to do useful calculations. I shall repeat only a few of the standard definitions here. With any random variable x 2 we can associate the distribution function F = Fx defined by F (t) = P[x ≤ t] (t ∈ R).
Some random variables admit a density function, i.e. a Lebesgue-integrable function f = fx satisfying
P[x ∈ B] =
B
f (t) dt
for all Borel sets B. As mentioned earlier, the integral
Ω x^ dP^ of an integrable random variable is denoted^ E[x] and is called the expectation, or mean, of x. (Integrable random variables are often referred to as having “finite mean”.) A useful formula for calculating the expectation of a non-negative random variable is
E[x] =
0
P[x > t] dt.
When x has a density function f we have (for Borel-measurable functions h)
E[h(x)] =
−∞
h(t)f (t) dt.
A random variable in L 2 (Ω) is said to have “finite variance” and we define var x = E[x^2 ] − E[x]^2 = E
[
(x − E[x])^2
]
An important example is the standard normal distribution N (0, 1), where the density function is
φ(t) =
2 π
e−^
1 2 t 2
(^2) Notice that we do not use upper case letters for random variables in this course.
