Example: The Schwarz space S( R

1.2. Fr´echet spaces Let us first recall some standard notations. For anyx= (x₁, . . . , x_d)2R^d

and ↵= (↵₁, . . . ,↵_d)2N^d0 one defines x^↵:=x^↵₁¹· · ·x^↵_d^d. For any 2N^d0, the symbol D denotes the partial derivative of order | | where | | := P_d

i=1 i, i.e.

D := @^{| |}

@x₁¹· · ·@x_d^d = @ ¹

@x₁¹ · · · @ ^d

@x_d^d.

Example: C

(⌦) with ⌦ ✓ R

open.

Let ⌦ ✓ R^d open in the euclidean topology. For any s 2 N⁰, we denote by C^s(⌦) the set of all real valued s times continuously di↵erentiable functions on ⌦, i.e. all the derivatives of order sexist (at every point of ⌦) and are continuous functions in ⌦. Clearly, when s = 0 we get the set C(⌦) of all real valued continuous functions on ⌦ and whens =1 we get the so-called set of all infinitely di↵erentiable functions orsmooth functions on ⌦. For any s2N⁰, C^s(⌦) (with pointwise addition and scalar multiplication) is a vector space over R.

Let us consider the following familyP of seminorms on C^s(⌦):

pm,K(f) := sup

2Nd

| |0m

sup

x2K

(D f)(x) ,8K⇢⌦ compact,8m2{0,1, . . . , s},

(Note when s= 1 we have m 2 N⁰.) The topology ⌧_P generated by P is usually referred asC^s-topology ortopology of uniform convergence on compact sets of the functions and their derivatives up to order s.

1) The C^s-topology clearly turns C^s(⌦) into a locally convex t.v.s., which is evidently Hausdor↵as the familyP is separating (see Prop 4.3.3 TVS-I). In- deed, if pm,K(f) = 0, 8m2{0,1, . . . , s} and8K compact subset of⌦then in particular p_0,{x}(f) =|f(x)|= 0 8x2⌦, which impliesf ⌘0 on⌦.

2) (C^s(⌦),⌧_P) is metrizable.

By Proposition 1.1.5, this is equivalent to prove that the C^s-topology can be generated by a countable separating family of seminorms. In order to show this, let us first observe that for any two non-negative integers m₁ m₂ s and any two compactK₁ ✓K₂ ⇢⌦we have:

p_m1,K1(f)p_m2,K2(f), 8f 2C^s(⌦).

Then the family{p_s,K :K ⇢⌦compact}generates theC^s topology onC^s(⌦).

Moreover, it is easy to show that there is a sequence of compact subsets {Kj}_j2N of ⌦ such that Kj ✓ K˚j+1 for all j 2 N and ⌦ = [_j2NKj. Then

9

1. Special classes of topological vector spaces

for any K ⇢ ⌦ compact we have that there exists j 2 N s.t. K ✓ K_j and sop_s,K(f)p_s,Kj(f),8f 2C^s(⌦). Hence, the countable family of seminorms {p_s,Kj :j 2N} generates the C^s topology on C^s(⌦) and it is separating. In- deed, if p_s,Kj(f) = 0 for all j 2N then for everyx 2⌦ we have x 2 K_i for somei2Nand so 0|f(x)|p_s,Ki(f) = 0, which implies |f(x)|= 0 for all x2⌦, i.e. f ⌘0 on⌦.

3)(C^s(⌦),⌧_P) is complete.

By Proposition 1.1.6, it is enough to show that it is sequentially complete.

Let (f_⌫)_⌫2N be a Cauchy sequence in C^k(⌦), i.e.

8ms,8K ⇢⌦ compact,8">0,9N 2Ns.t. 8µ,⌫ N : p_m,K(f_⌫ f_µ)".

(1.7) In particular, for any x 2⌦ by taking m = 0 and K ={x} we get that the sequence (f_⌫(x))_⌫2N is a Cauchy sequence in R. Hence, by the completeness of R, it has a limit point inRwhich we denote byf(x). Obviouslyx7!f(x) is a function on⌦, so we have just showed that the sequence (f_⌫)_⌫2Nconverge tof pointwise in ⌦, i.e.

8x2⌦,8">0,9M_x2Ns.t. 8µ M_x: |f_µ(x) f(x)|". (1.8) Then it is easy to see that (f_⌫)_⌫2N converges uniformly tof in every compact subsetK of⌦. Indeed, we get it just passing to the pointwise limit forµ! 1 in (1.7) for m= 0. ²

As (f⌫)⌫2N converges uniformly to f in every compact subset K of ⌦, by taking this subset identical with a suitable neighbourhood of any point of ⌦, we conclude by Lemma 1.2.2thatf is continuous in ⌦.

• If s = 0, this completes the proof since we just showed f⌫ ! f in the C⁰ topology andf 2C(⌦).

• If 0 < s < 1, then observe that since (f⌫)⌫2N is a Cauchy sequence in C^s(⌦), for each j 2 {1, . . . , d} the sequence (_@x^@

jf_⌫)_⌫2N is a Cauchy sequence in C^{s 1}(⌦). Then induction on s allows us to conclude that, for each j 2 {1, . . . , d}, the (_@x^@

jf⌫)⌫2N converges uniformly on every compact subset of ⌦ to a function g^(j) 2C^{s 1}(⌦) and by Lemma 1.2.3 we have thatg^(j) = _@x^@

jf. Hence, we have showed that (f⌫)⌫2Nconverges tof in theC^s topology with f 2C^s(⌦).

2Detailed proof: Let">0. By (1.7) form= 0,9N2Ns.t.8µ,⌫ N : |f⌫(x) fµ(x)|

"

2,8x2K.Now for each fixedx2K one can always choose aµx larger than both N and the correspondingMx as in (1.8) so that|fµx(x) f(x)|^"2. Hence, for all⌫ N one gets

that|f⌫(x) f(x)||f⌫(x) fµx(x)|+|fµx(x) f(x)|",8x2K

10

1.2. Fr´echet spaces

• Ifs=1, then we are also done by the definition of theC¹-topology. In- deed, a Cauchy sequence (f_⌫)_⌫2N inC¹(⌦) it is in particular a Cauchy sequence in the subspace topology given by C^s(⌦) for any s 2 N and hence, for what we have already showed, it converges to f 2 C^s(⌦) in the C^s topology for any s2N. This means exactly that (f_⌫)_⌫2N con- verges tof 2C¹(⌦) in the in C¹ topology.

Let us prove now the two lemmas which we have used in the previous proof:

Lemma 1.2.2. Let A ⇢ R^d and (f_⌫)_⌫2N in C(A). If (f_⌫)_⌫2N converges to a function f uniformly in A then f 2C(A).

Proof.

Let x₀ 2 A and "> 0. By the uniform convergence of (f_⌫)_⌫2N tof in A we get that:

9N 2Ns.t. 8⌫ N :|f⌫(y) f(y)| "

3,8y2A.

Fix such a ⌫. Asf⌫ is continuous onAthen:

9 >0 s.t. 8x2Awith |x x₀| we have |f_⌫(x) f_⌫(x₀)| "

3. Therefore, we obtain that 8x2A with|x x0| :

|f(x) f(x₀)||f(x) f_⌫(x)|+|f_⌫(x) f_⌫(x₀)|+|f_⌫(x₀) f(x₀)|".

Lemma 1.2.3. Let A⇢R^d and (f_⌫)_⌫2N in C¹(A). If (f_⌫)_⌫2N converges to a functionf uniformly inA and for eachj2{1, . . . , d} the sequence(_@x^@

jf_⌫)_⌫2N converges to a function g^(j) uniformly in A, then

g^(j)= @

@xj

f, 8j2{1, . . . , d}. This means in particular that f 2C¹(A).

Proof. (ford= 1, A= [a, b])

By the fundamental theorem of calculus, we have that for any x2A f_⌫(x) f_⌫(a) =

Z _x

a

@

@tf_⌫(t)dt. (1.9)

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1. Special classes of topological vector spaces

By the uniform convergence of the first derivatives tog⁽¹⁾and by the Lebesgue dominated convergence theorem, we also have

Z x

a

@

@tf_⌫(t)dt! Z x

a

g⁽¹⁾(t)dt, as⌫! 1. (1.10) Using (1.9) and (1.10) together with the assumption thatf⌫ !f unformly in A, we obtain that:

f(x) f(a) = Z x

a

g⁽¹⁾(t)dt, i.e. _@x^@ f (x) =g⁽¹⁾(x),8x2A.

Example: The Schwarz space S( R

).

The Schwartz space or space of rapidly decreasing functions on R^d is defined as the set S(R^d) of all real-valued functions which are defined and infinitely di↵erentiable on R^d 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.

S(R^d) :=

(

f 2C¹(R^d) : sup

x2R^d

x^↵(D f)(x) <1, 8↵, 2N^d0

) . (For example, any smooth functionf with compact support inR^dis inS(R^d), since any derivative of f is continuous and supported on a compact subset of R^d, sox^↵(D f(x)) has a maximum inR^d by the extreme value theorem.)

The Schwartz space S(R^d) is a vector space overR and we equip it with the topology ⌧_Q given by the family Q of seminorms onS(R^d):

q_m,k(f) := sup

2Nd

| |m0

sup

x2R^d

(1 +|x|)^k (D )f(x) , 8m, k2N0.

Note thatf 2S(R^d) if and only if8m, k 2N⁰, q_m,k(f)<1.

The space S(R^d) is a linear subspace of C¹(R^d), but ⌧_Q is finer than the subspace topology induced on it by ⌧_P where P is the family of seminorms defined on C¹(R^d) as in the above example. Indeed, it is clear that for any f 2S(R^d), any m2N⁰ and anyK ⇢R^dcompact we havepm,K(f)qm,0(f) which gives the desired inclusion of topologies.

1)(S(R^d),⌧_Q) is a locally convex t.v.s. which is also evidently Hausdor↵since the family Q is separating. Indeed, if q_m,k(f) = 0, 8m, k 2 N⁰ then in particular q_0,0(f) = sup_x2Rd|f(x)|= 0, which impliesf ⌘0 onR^d.

2) (S(R^d),⌧_Q) is a metrizable, as Q is countable and separating (see Propo- sition 1.1.5).

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