1.2. Fr´echet spaces 1.2 Fr´echet spaces Definition 1.2.1. A complete metrizable locally convex t.v.s. is called a Fr´echet space (or F-space) Note that by Theorem 1.1 and Proposition 1.1.8, any Fr´echet space is in particular a HausdorffBaire space. Combining the properties of metrizable t.v.s. which we proved in Sheet 1 and the results about complete t.v.s. which we have seen in TVS-I, we easily get the following properties: Any closed subspace of an F-space endowed with the induced topology • is an F-space. The product of a countable family of F-spaces is an F-space. • The quotient of an F-space modulo a closed subspace is an F-space. • Examples of F-spaces are: Hausdorfffinite dimensional t.v.s., Hilbert spaces, and Banach spaces. In the following we will present two examples of F-spaces which do not enter in any of the latter categories. d Let us first recall some standard notations. For any x =(x1,...,xd) R d α α1 αd d∈ and α =(α1,...,αd) N one defines x := x x . For any β N ,the ∈ 0 1 ··· d ∈ 0 symbol Dβ denotes the partial derivative of order β where β := d β , | | | | i=1 i i.e. ∂ β ∂β1 ∂βd Dβ := | | = . ∂xβ1 ∂xβd ∂xβ1 ···∂xβd 1 ··· d 1 d Example: k(Ω) with Ω Rd open. dC ⊆ Let Ω R open in the euclidean topology. For any k N0, we denote by ⊆ ∈ k(Ω) the set of all real valued k times continuously differentiable functions C − on Ω, i.e. all the derivatives of f of order k exist (at every point of Ω) and ≤ are continuous functions in Ω. Clearly, when k = 0 we get the set (Ω) of all C real valued continuous functions on Ωand when k = we get the so-called ∞ set of all infinitely differentiable functions or smooth functions on Ω. For any k k N0, (Ω) (with pointwise addition and scalar multiplication) is a vector ∈ C space over R. Let us consider the following family of seminorms on k(Ω): P C β pm,K (f):= sup sup (D f)(x) , K Ωcompact, m 0, 1,...,k , β d x K ∀ ⊂ ∀ ∈{ } ∈N0 ∈ β m | |≤ (Note when k = we have m N0.) The topology τ generated by is usually referred as∞k-topology or ∈topology of uniform convergenceP on compactP C sets of the functions and their derivatives up to order k. 9 1. Special classes of topological vector spaces 1) The k-topology clearly turns k(Ω) into a locally convex t.v.s.,whichis C C evidently Hausdorff as the family is separating (see Prop 4.3.3 TVS-I). In- P deed, if p (f) = 0, m 0, 1,...,k and K compact subset of Ωthen in m,K ∀ ∈{ } ∀ particular p0, x (f)= f(x) =0 x Ω, which implies f 0 on Ω. { } | | ∀ ∈ ≡ 2) ( k(Ω),τ ) is metrizable. C P By Proposition 1.1.5, this is equivalent to prove that the k-topology can be C generated by a countable family of seminorms. In order to show this, let us first observe that for any two non-negative integers m m and any two 1 ≤ 2 compact K K Ωwehave: 1 ⊆ 2 ⊂ p (f) p (f), f k(Ω). m1,K1 ≤ m2,K2 ∀ ∈C Then the family p : K Ωcompact generates the k topology on { k,K ⊂ } C − k(Ω). Moreover, it is easy to show that there is a sequence of compact sub- C ˚ sets Kj j N of Ωsuch that Kj Kj+1 for all j N and Ω= j NKj.Then { } ∈ ⊆ ∈ ∪ ∈ for any K Ωcompact we have that there exists j N s.t. K Kj and so ⊂ k ∈ ⊆ pk,K(f) pk,Kj (f), f (Ω). Hence, the countable family of seminorms ≤ ∀ ∈C k k pk,Kj : j N generates the topology on (Ω). { ∈ } C − C 3) ( k(Ω),τ ) is complete. C P By Proposition 1.1.6, it is enough to show that it is sequentially complete. k Let (fν)ν N be a Cauchy sequence in (Ω), i.e. ∈ C m k, K Ωcompact, ε>0, N N s.t. µ, ν N : pm,K (fν fµ) ε. ∀ ≤ ∀ ⊂ ∀ ∃ ∈ ∀ ≥ − ≤ (1.5) In particular, for any x Ωby taking m = 0 and K = x we get that the ∈ { } sequence (fν(x))ν N is a Cauchy sequence in R. Hence, by the completeness ∈ of R, it has a limit point in R which we denote by f(x). Obviously x f(x) → is a function on Ω, so we have just showed that the sequence (fν)ν N converge ∈ to f pointwise in Ω, i.e. x Ω, ε>0, Mx N s.t. µ Mx : fµ(x) f(x) ε. (1.6) ∀ ∈ ∀ ∃ ∈ ∀ ≥ | − |≤ Then it is easy to see that (fν)ν N converges uniformly to f in every compact ∈ subset K of Ω. Indeed, we get it just passing to the pointwise limit for µ →∞ in (1.5) for m = 0. 1 1 Detailed proof:Letε>0. By (1.5) for m =0, N N s.t. µ, ν N : fν (x) fµ(x) ε ∃ ∈ ∀ ≥ | − |≤ 2 , x K. Now for each fixed x K one can always choose a µx larger than both N and ∀ ∈ ∈ ε the corresponding Mx as in (1.6)sothat fµ (x) f(x) . Hence, for all ν N one gets | x − |≤ 2 ≥ that fν (x) f(x) fν (x) fµ (x) + fµ (x) f(x) ε, x K | − |≤| − x | | x − |≤ ∀ ∈ 10 1.2. Fr´echet spaces As, (fν)ν N converges uniformly to f in every compact subset K of Ω, by ∈ taking this subset identical with a suitable neighborhood of any point of Ω, we conclude by Lemma 1.2.2 that f is continuous in Ω. If k = 0, this completes the proof since we just showed f f in the • ν → 0 topology and f (Ω). C − ∈C If 0 <k< , then observe that since (fν)ν N is a Cauchy sequence • k ∞ ∈ ∂ in (Ω), for each j 1,...,d the sequence ( fν)ν N is a Cauchy C ∈{ } ∂xj ∈ k 1 sequence in − (Ω). Then induction on k allows us to conclude that, C ∂ for each j 1,...,d ,the( fν)ν N converges uniformly on every ∈{ } ∂xj ∈ (j) k 1 compact subset of Ωto a function g − (Ω) and by Lemma 1.2.3 (j) ∂ ∈C we have that g = f. Hence, we have showed that (fν)ν N converges ∂xj ∈ to f in the k topology with f k(Ω). C − ∈C If k = , then we are also done by the definition of the -topology. • ∞ C∞ Indeed, a Cauchy sequence (fν)ν N in ∞(Ω) it is in particular a Cauchy ∈ C sequence in the subspace topology given by k(Ω) for any k N and C ∈ hence, for what we have already shown, it converges to f k(Ω) in the k ∈C C topology for any k N. This means exactly that (fν)ν N converges − ∈ ∈ to f (Ω) in the in topology. ∈C∞ C∞− Let us prove now the two lemmas which we have used in the previous proof: d Lemma 1.2.2. Let A R and (fν)ν N in (A).If(fν)ν N converges to a ⊂ ∈ C ∈ function f uniformly in A then f (A). ∈C Proof. Let x A and ε>0. By the uniform convergence in A we get that: 0 ∈ ε N N s.t. ν N : fν(y) f(y) , y A. ∃ ∈ ∀ ≥ | − |≤3 ∀ ∈ Fix such a ν. As fν is continuous on A then: ε δ>0s.t. x A with x x δ : f (x) f (x ) . ∃ ∀ ∈ | − 0|≤ | ν − ν 0 |≤3 Therefore, we obtain that x A with x x δ : ∀ ∈ | − 0|≤ f(x) f(x ) f(x) f (x) + f (x) f (x ) + f (x ) f(x ) ε. | − 0 |≤| − ν | | ν − ν 0 | | ν 0 − 0 |≤ 11 1. Special classes of topological vector spaces d 1 Lemma 1.2.3. Let A R and (fν)ν N in (A).If(fν)ν N converges to a ⊂ ∈ C ∈ ∂ function f uniformly in A and for each j 1,...,d the sequence ( fν)ν N ∈{ } ∂xj ∈ converges to a function g(j) uniformly in A, then ∂ g(j) = f, j 1,...,d . ∂xj ∀ ∈{ } This means in particular that f 1(A). ∈C Proof. (for d = 1, A =[a, b]) By the fundamental theorem of calculus, we have that for any x A ∈ x f (x) f (a)= f (t)dt. ν − ν ν a By the uniform convergence of the derivatives and the dominated convergence theorem, we also have x x (1) f (t)dt g (t)dt, as ν . ν → →∞ a a Thus x f(x) f(a)= g(1)(t)dt, − a i.e. f (x)=g(1)(x), x A. ∀ ∈ Example: The Schwarz space S(Rd). The Schwartz space or space of rapidly decreasing functions on Rd is defined as the set (Rd) of all real-valued functions which are defined and infinitely S differentiable on Rd and which have the additional property (regulating their growth at infinity) that all their derivatives tend to zero at infinity faster than any inverse power of x,i.e. d d α β d (R ):= f ∞(R ): sup x (D f)(x) < , α,β N0 . S ∈C x Rd ∞ ∀ ∈ ∈ (For example, any smooth function f with compact support in Rd is in (Rd), S since any derivative of f is continuous and supported on a compact subset of Rd,soxα(Dβf(x)) has a maximum in Rd by the extreme value theorem.) 12 1.2.