B.1 §B. Appendix B. Topological vector spaces B.1. Fr´echet spaces. In this appendix we go through the definition of Fr´echet spaces and their inductive limits, such as they are used for definitions of function spaces in Chapter 2. A reader who just wants an orientation about Fr´echet spaces will not have to read every detail, but need only consider Definition B.4, Theorem B.5, Remark B.6, Lemma B.7, Remark B.7a and Theorem B.8, skipping the proofs of Theorems B.5 and B.8. For inductive limits of such spaces, Section B.2 gives an overview (the proofs are established in a succession of exercises), ∞ and all that is needed for the definition of the space C0 (Ω) is collected in Theorem 2.5. We recall that a topological space S is a space provided with a collection τ of subsets (called the open sets), satisfying the rules: S is open, ∅ is open, the intersection of two open sets is open, the union of any collection of open sets is open. The closed sets are then the complements of the open sets. A neighborhood of x ∈ S is a set containing an open set containing x. Recall also that when S and S1 are topological spaces, and f is a mapping from S to S1, then f is continuous at x ∈ S when there for any neighborhood V of f(x) in S1 exists a neighborhood U of x in S such that f(U) ⊂ V . Much of the following material is also found in the book of Rudin [R 1974], which was an inspiration for the formulations here. Definition B.1. A topological vector space (t.v.s.) over the scalar field L = R or C (we most often consider C), is a vector space X provided with a topology τ having the following properties: (i) A set consisting of one point {x} is closed. (ii) The maps {x,y} 7→ x + y from X × X into X (B.1) {λ, x} 7→ λx from L × X into X are continuous. In this way, X is in particular a Hausdorff space, cf. Exercise B.4. (We here follow the terminology of [R 1974] and [P 1989] where (i) is included in the definition of a t.v.s.; all spaces that we shall meet have this property. In part of the literature, (i) is not included in the definition and one speaks of Hausdorff topological vector spaces when it holds.) L is considered with the usual topology for R or C. B.2 Definition B.2. A set Y ⊂ X is said to be a) convex, when y1,y2 ∈ Y and t ∈ ]0, 1[ imply ty1 + (1 − t)y2 ∈ Y , b) balanced, when y ∈ Y and |λ| ≤ 1 imply λy ∈ Y ; c) bounded (with respect to τ), when there for every neighborhood U of 0 exists t > 0 so that Y ⊂ tU. Note that boundedness is defined without reference to “balls” or the like. Lemma B.3. Let X be a topological vector space. 1◦ Let a ∈ X and λ ∈ L \{0}. The maps from X to X Ta : x 7→ x − a (B.2) Mλ : x 7→ λx are continuous with continuous inverses T−a resp. M1/λ. 2◦ For any neighborhood V of 0 there exists a balanced neighborhood W of 0 such that W + W ⊂ V . 3◦ For any convex neighborhood V of 0 there exists a convex balanced neighborhood W of 0 so that W ⊂ V . Proof. 1◦ follows directly from the definition of a topological vector space. For 2◦ we appeal to the continuity of the two maps in (B.2) as follows: Since {x,y} 7→ x + y is continuous at {0, 0}, there exist neighborhoods W1 and W2 of 0 so that W1 + W2 ⊂ V . Since {λ, x} 7→ λx is continuous at {0, 0} there exist balls B(0,r1) and B(0,r2) in L with r1,r2 > 0 and neighborhoods 0 0 W1 and W2 of 0 in X, such that 0 0 B(0,r1)W1 ⊂ W1 and B(0,r2)W2 ⊂ W2. 0 0 Let r = min{r1,r2} and let W = B(0,r)(W1 ∩ W2), then W is a balanced neighborhood of 0 with W + W ⊂ V . ◦ ◦ For 3 we first choose W1 as under 2 , so that W1 is a balanced neigh- borhood of 0 with W1 ⊂ V . Let W = α∈L,|α|=1 αV . Since W1 is balanced, −1 α W1 = W1 for all |α| = 1, hence W1T⊂ W . Thus W is a neighborhood of 0. It is convex, as an intersection of convex sets. That W is balanced is seen as follows: For λ = 0, λW ⊂ W . For 0 < |λ| ≤ 1, λ λW = |λ| αV = |λ| βV ⊂ W |λ| α∈L\,|α|=1 β∈L\,|β|=1 B.3 (with β = αλ/|λ|); the last inclusion follows from the convexity since 0 ∈ W . Note that in 3◦ the interior W ◦ of W is an open convex balanced neigh- borhood of x. The lemma implies that the topology of a t.v.s. X is translation invariant, i.e., E ∈ τ ⇐⇒ a + E ∈ τ for all a ∈ X. The topology is therefore determined from the system of neighborhoods of 0. Here it suffices to know a local basis for the neighborhood system at 0, i.e., a system B of neighborhoods of 0 such that every neighborhood of 0 contains a set U ∈ B. X is said to be locally convex, when it has a local basis of neighborhoods at 0 consisting of convex sets. X is said to be metrizable, when it has a metric d such that the topology on X is identical with the topology defined by this metric; this happens 1 exactly when the balls B(x, n ), n ∈ N, are a local basis for the system of neighborhoods at x for any x ∈ X. Here B(x,r) denotes as usual the open ball B(x,r) = {y ∈ X | d(x,y) <r} , and we shall also use the notation B(x,r) = {y ∈ X | d(x,y) ≤ r} for the closed ball. Such a metric need not be translation invariant, but it will usually be so in the cases we consider; translation invariance (also just called invariance) here means that d(x + a,y + a) = d(x,y) for x,y,a ∈ X . One can show, see e.g. [R 1974, Th. 1.24], that when a t.v.s. is metrizable, then the metric can be chosen to be translation invariant. A Cauchy sequence in a t.v.s. X is a sequence (xn)n∈N with the property: For any neighborhood U of 0 there exists an N ∈ N so that xn − xm ∈ U for n and m ≥ N. In a metric space (M,d), Cauchy sequences — let us here call them met- ric Cauchy sequences — are usually defined as sequences (xn) for which d(xn,xm) → 0 in R for n and m →∞. This property need not be preserved if the metric is replaced by another equivalent metric (defining the same topology). We have however for t.v.s. that if the topology in X is given by an invariant metric d, then the general concept of Cauchy sequences for X gives just the metric Cauchy sequences (Exercise B.2). A metric space is called complete, when every metric Cauchy sequence is convergent. More generally we call a t.v.s. sequentially complete, when every Cauchy sequence is convergent. Banach spaces and Hilbert spaces are of course complete metrizable topo- logical vector spaces. The following more general type is also important: B.4 Definition B.4. A topological vector space is called a Fr´echet space, when X is metrizable with a translation invariant metric, is complete, and is locally convex. The local convexity is mentioned explicitly because the balls belonging to a given metric need not be convex, cf. Exercise B.1. (One has however that if X is metrizable and locally convex, then there exists a metric for X with convex balls, cf. [R 1974, Th. 1.24].) Note that the balls defined from a norm are convex. It is also possible to define Fr´echet space topologies (and other locally convex topologies) by use of seminorms; we shall now take a closer look at this method to define topologies. Recall that a seminorm on a vector space X is a function p : X → R+ with the properties (i) p(x + y) ≤ p(x) + p(y) for x,y ∈ X (subadditivity) , (B.3) (ii) p(λx) = |λ|p(x) for λ ∈ L and x ∈ X (multiplicativity) . A family P of seminorms is called separating, when for every x0 ∈ X \{0} there is a p ∈ P such that p(x0) > 0. Theorem B.5. Let X be a vector space and let P be a separating family of seminorms on X. Define a topology on X by taking, as a local basis B for the system of neighborhoods at 0, the convex balanced sets V (p, ε) = {x | p(x) < ε} , p ∈ P and ε > 0 , (B.4) together with their finite intersections W (p1,...,pN ; ε1,...,εN ) = V (p1, ε1) ∩···∩ V (pN , εN ); (B.5) and letting a local basis for the system of neighborhoods at each x ∈ X consist of the translated sets x + W (p1,...,pN ; ε1,...,εN ). (It suffices to let εj = 1/nj, nj ∈ N.) With this topology, X is a topological vector space. The seminorms p ∈ P are continuous maps of X into R.