A.7 Some functional analysis

Numerous arguments in this book involve infinite-dimensional vector spaces. Typical examples arising in connection with a probability space (Ω,F, P) are the spaces Lp := Lp(Ω,F, P) for 0 ≤ p ≤ ∞, which we will introduce below. To this end, we first take p ∈ (0,∞] and denote by Lp(Ω,F, P) the set of all F-measurable functions Z on (Ω,F, P) such that Zp < ∞, where

Let us also introduce the space L0(Ω,F, P), defined as the set of all P-a.s. finite random variables. If no ambiguity with respect to σ-algebra and measure can arise, we may sometimes write Lp(P) or just Lp instead of Lp(Ω,F, P). For p ∈ [0,∞], the space Lp(Ω,F, P), or just Lp, is obtained from Lp by identifying random variables which coincide up to a P-null set. Thus, Lp consists of all equivalence classes with respect to the equivalence relation

If p ∈ [1,∞] then the vector space Lp is a Banach space with respect to the norm · p defined in (A.34), i.e., every Cauchy sequence with respect to · p converges to some element in Lp. In principle, one should distinguish between a random variable Z ∈ Lp and its associated equivalence class [Z] ∈ Lp, of which Z is a representative element. In order to keep things simple, we will follow the usual convention of identifying Z with its equivalence class, i.e., we will just write Z ∈ Lp.

On the space L0, we use the topology of convergence in P-measure. This topology is generated by the metric

Note, however, that d is not a norm.

Definition A.57. A linear space E which carries a topology is called a topological vector space if every singleton {x} for x ∈ E is a closed set, and if the vector space operations are continuous in the following sense:

is a continuous mapping from E × E into E, and

is a continuous mapping from ℝ × E into E.

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Clearly, every Banach space is a topological vector space. The following result is a generalization of the separation argument in Proposition A.5 to an infinite-dimensional setting.

Theorem A.58. In a topological vector space E, any two disjoint convex sets B and C , one of which has an interior point, can be separated by a nonzero continuous linear functional on E, i.e.,

Proof. See [104], Theorem V.2.8.

If one wishes to strictly separate two convex sets by a linear functional in the sense that one has a strict inequality in (A.37), then one needs additional conditions both on the convex sets and on the underlying space E.

Definition A.59. A topological vector space E is called a locally convex space if its topology has a base consisting of convex sets.

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If E is a Banach space with norm · , then the open balls

form by definition a base for the topology of E. Since such balls are convex sets, any Banach space is locally convex. The space L0(Ω,F, P) with the topology of convergence in P-measure, however, is not locally convex if (Ω,F, P) has no atoms; see, e.g., Theorem 12.41 of [3].

The following theorem is one variant of the classical Hahn–Banach theorem on the existence of “separating hyperplanes”.

Theorem A.60 (Hahn–Banach). Suppose that Band C are two nonempty, disjoint, and convex subsets of a locally convex space E. Then, if B is compact and C is closed, there exists a continuous linear functional on E such that

Proof. See, for instance, [247], p. 65, or [104], Theorem V.2.10.

One corollary of the preceding result is that, on a locally convex space E, the collection

separates the points of E, i.e., for any two distinct points x, y ∈ E there exists some ∈ Eʹ such that (x) ≠ (y). The space Eʹ is called the dual or the dual space of E. For instance, if p ∈ [1,∞) it is well-known that the dual of Lp := Lp(Ω,F, P) is given by Lq, where More precisely, for every ∈ (Lp)ʹ there exists Y ∈ Lq such that

see, e.g., Theorems IV.8.1 and IV.8.5 in [104]. The following definition describes a natural way in which locally convex topologies often arise.

Definition A.61. Let E be linear space, and suppose that F is a linear class of linear functionals on E which separates the points of E. The F-topology on E, denoted by σ(E, F), is the topology on E which is obtained by taking as a base all sets of the form

where n ∈ ℕ, x ∈ E, i ∈ F, and r > 0. If E already carries a locally convex topology, then the E ʹ-topology σ(E, E ʹ) is called the weak topology on E.

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If E is infinite-dimensional, then E is typically not metrizable in the F-topology. In this case, it may not suffice to consider converging sequences when making topological assertions; see, however, Theorem A.69 below. The following proposition summarizes a few elementary properties of the F-topology.

Proposition A.62. Consider the situation of the preceding definition. Then:

(a) E is a locally convex space for the F-topology.

(b) The F-topology is the coarsest topology on E for which every ∈ F is continuous.

(c) The dual of E for the F-topology is equal to F.

Proof. See, e.g., Section V.3 of [104].

Theorem A.63. Suppose that E is a locally convex space and that C is a convex subset of E. Then C is weakly closed if and only if C is closed in the original topology of E.

Proof. If the convex set C is closed in the original topology then, by Theorem A.60, it is equal to the intersection of the halfspaces H = { ≤ c} such that H ⊃ C , and thus closed in the weak topology σ(E, Eʹ). The converse is clear.

For a given locally convex space E we can turn things around and consider E as a set of linear functionals on the dual space Eʹ by letting x() := (x) for ∈ Eʹ and x ∈ E. The E-topology σ(Eʹ , E) obtained in this way is called the weak∗ topology on Eʹ. According to part (c) of Proposition A.62, E is then the topological dual of (Eʹ , σ(Eʹ , E)). For example, the Banach space L∞ := L∞(Ω,F, P) is the dual of L1, but the converse is generally not true; see our discussion on finitely additive set functions in Section A.6 and, for a precise statement, Theorem IV.8.16 in [104]. However, L1 becomes the dual of L∞ if we endow L∞ with the weak∗ topology σ(L∞, L1).

The mutual duality between E and Eʹ allows us to state a general version of part (b) of Proposition A.9.

Definition A.64. The Fenchel–Legendre transform of a function f : E → ℝ ∪ {+∞} is the function f ∗ on Eʹ defined by

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If f ≡ +∞, then f ∗ is a proper convex and lower semicontinuous function as the supremum of affine functions. If f is itself a proper convex function, then f ∗ is also called the conjugate function of f .

Theorem A.65. Let f be a proper convex function on a locally convex space E. If f is lower semicontinuous with respect to σ(E, Eʹ), then f = f ∗∗.

It is straightforward to adapt the proof we gave in the one-dimensional case of Proposition A.9 to the infinite-dimensional situation of Theorem A.65; all one has to do is to replace the separating hyperplane lemma by the Hahn–Banach separation theorem in the form of Theorem A.60.

One of the reasons for considering the weak topology on a Banach space or, more generally, on a locally convex space is that typically more sets are compact for the weak topology than for the original topology. The following result shows that the unit ball in the dual of a Banach space is weak∗ compact. Here we use the fact that a Banach space (E, · E) defines the following norm on its dual Eʹ:

Theorem A.66 (Banach–Alaoglu). Let E be a Banach space with dual Eʹ. Then { ∈ Eʹ | E≤ r} is weak∗ compact for every r ≥ 0.

Proof. See, e.g., Theorem IV.21 in [230].

Theorem A.67 (Krein–Šmulian). Let E be a Banach space and suppose that C is a convex subset of the dual space Eʹ. Then C is weak∗ closed if and only if

is weak∗ closed for each r > 0.

Proof. See Theorem V.5.7 in [104].

The preceding theorem implies the following characterization of weak∗ closed sets in L∞ = L∞(Ω,F, P) for a given probability space (Ω,F, P).

Lemma A.68. A convex subset C of L∞ is weak∗ closed if for every r > 0

is closed in L1.

Proof. Since Cr is convex and closed in L1, it is weakly closed in L1 by Theorem A.63. Since the natural injection

is continuous, Cr is σ(L∞, L1)-closed in L∞. Thus, C is weak∗ closed due to the Krein–Šmulian theorem, applied with E = L1 and Eʹ = L∞.

Finally, we state a few fundamental results on weakly compact sets.

Theorem A.69 (Eberlein–Šmulian). For any subset A of a Banach space E, the following conditions are equivalent:

(a) A is weakly sequentially compact, i.e., any sequence in A has a subsequence which converges weakly in E.

(b) A is weakly relatively compact, i.e., the weak closure of A is weakly compact.

Proof. See [104], Theorem V.6.1.

The following result characterizes the weakly relatively compact subsets of the Banach space L1 := L1(Ω,F, P). It implies, in particular, that a set of the form {f ∈ L1 | |f | ≤ g} with given g ∈ L1 is weakly compact in L1.

Theorem A.70 (Dunford–Pettis). A subset A of L1 is weakly relatively compact if and only if it is bounded and uniformly integrable.

Proof. See, e.g., Theorem IV.8.9 or Corollary IV.8.11 in [104].