1.3. Inductive topologies and LF-spaces neighbourhood of the origin in (E, ⌧

neighbourhood of the origin in (E,⌧_ind) sinceU\E_mis a neighbourhood of the origin in (E_m,⌧_m) for all m 2N. We can then conclude that⌧_n✓⌧_ind E_n, 8n2N.

From the previous proposition we can easily deduce that any LF-space is not only a locally convex t.v.s. but also Hausdor↵. Indeed, if (E,⌧_ind) is the LF-space with defining sequence {(E_n,⌧_n) : n 2 N} and we denote by F(o) [resp. Fn(o)] the filter of neighbourhoods of the origin in (E,⌧_ind) [resp. in (E_n,⌧_n)], then:

\

V2F(o)

V = \

V2F(o)

V \ [

n2N

En

!

= [

n2N

\

V2F(o)

(V \En) = [

n2N

\

Un2Fn(o)

Un={o},

which implies that (E,⌧_ind) is Hausdor↵by Corollary 2.2.4 in TVS-I.

As a particular case of Proposition1.3.1 we get that:

Proposition 1.3.5.

Let (E,⌧_ind) be an LF-space with defining sequence {(E_n,⌧_n) : n 2 N} and (F,⌧) an arbitrary locally convex t.v.s..

1. A linear mappingu fromE intoF is continuous if and only if, for each n2N, the restriction u En of u to En is continuous.

2. A linear form on E is continuous if and only if its restrictions to each E_n are continuous.

Note that Propositions1.3.4and1.3.5hold for any countable strict induc- tive limit of an increasing sequences of locally convex Hausdor↵ t.v.s. (even when they are not Fr´echet).

The following result is instead typical of LF-spaces as it heavily relies on the completeness of the t.v.s. of the defining sequence. Before introducing it, let us introduce the concept of accumulation point for a filter of a topological space together with some basic useful properties.

Definition 1.3.6. LetF be a filter of a topological spaceX. A pointx2X is called an accumulation point of a filter F if x belongs to the closure of every set which belongs to F, i.e. x2M , 8M 2F.

Proposition 1.3.7. If a filterF of a topological spaceX converges to a point x, then x is an accumulation point of F.

Proof. Suppose that x were not an accumulation point of F. Then there would be a set M 2F such that x /2M. Hence, X\M is open in X and so it is a neighbourhood of x. Then X\M 2F asF ! x by assumption. But F is a filter and so M\ X\M 2F and so M \ X\M 6=;, which is a contradiction.

Proposition 1.3.8. If a Cauchy filter F of a t.v.s. X has an accumulation pointx, then F converges to x.

Proof. (Christmas assignment)

Theorem 1.3.9. Any LF-space is complete.

Proof.

Let (E,⌧_ind) be an LF-space with defining sequence{(E_n,⌧_n) :n2N}. LetF be a Cauchy filter on (E,⌧_ind). Denote by FE(o) the filter of neighbourhoods of the origin in (E,⌧_ind) and consider

G:={A✓E: A◆M+V for someM 2F, V 2FE(o)}. 1)G is a filter on E.

Indeed, it is clear from its definition that G does not contain the empty set and that any subset ofE containing a set inG has to belong to G. Moreover, for anyA₁, A₂ 2Gthere existM₁, M₂2F,V₁, V₂ 2FE(o) s.t. M₁+V₁ ✓A₁ and M2+V2 ✓A2; and therefore

A1\A2 ◆(M1+V1)\(M2+V2)◆(M1\M2) + (V1\V2).

The latter proves that A1\A22G sinceF and FE(o) are both filters and so M₁\M₂ 2F and V₁\V₂ 2FE(o).

2)G ✓F.

In fact, for anyA2G there existM 2F and V 2FE(o) s.t.

A◆M+V M +{0}=M which implies that A2F sinceF is a filter.

3)G is a Cauchy filter on E.

Let U 2 FE(o). Then there always exists V 2 FE(o) balanced such that V +V V ✓U. AsF is a Cauchy filter on (E,⌧_ind), there existsM 2F such that M M ✓V. Then

(M+V) (M+V)✓(M M) + (V V)✓V +V V ✓U which proves that G is a Cauchy filter since M+V 2G.

It is possible to show (and we do it later on) that:

9p2N: 8A2G, A\E_p 6=; (1.13) This property ensures that the family

Gp:={A\Ep: A2G}

is a filter on E_p. Moreover, since G is a Cauchy filter on (E,⌧_ind) and since by Proposition 1.3.4we have ⌧_ind Ep =⌧p,Gp is a Cauchy filter on (Ep,⌧p).

Hence, the completeness ofE_pguarantees that there existsx2E_ps.t. Gp !x which implies in turn that x is an accumulation point for Gp by Proposition 1.3.7. In particular, this gives that for any A 2 G we have x 2 A\Ep⌧p

✓ A\E_p^⌧indA^⌧ind, i.e. xis an accumulation point for the Cauchy filterG. Then, by Proposition 1.3.8, we get that G !x, and so FE(o)✓G✓F which gives F !x.

Proof. of (1.13)

Suppose that (1.13) is false, i.e. 8n2N, 9A_n 2G s.t. A_n\E_n =;. By the definition of G, this means that

8n2N,9M_n2F, V_n2FE(o),s.t. (M_n+V_n)\E_n=;. (1.14) Since E is a locally convex t.v.s., we may assume that each V_n is balanced and convex, and that V_n+1✓V_n for all n2N. Consider

W_n:=conv V_n[

n[1 k=1

(V_k\E_k)

! ,

then

(W_n+M_n)\E_n=;,8n2N.

Indeed, if there existsh2(Wn+Mn)\Enthenh2Enandh2(Wn+Mn). We may then write: h=x+ty+ (1 t)zwithx2M_n,y2V_n,z2V₁\E_{n 1}and t2[0,1]. Hence,x+ty=h (1 t)z2E_n. But we also havex+ty2M_n+V_n, sinceVnis balanced and soty2Vn. Therefore,x+ty2(Mn+Vn)\Enwhich contradicts (1.14).

Now let us define

W :=conv [1 k=1

(V_k\E_k)

! .

As W is convex and as W \E_k contains V_k \E_k for all k 2 N, W is a neighbourhood of the origin in (E,⌧_ind). Moreover, as (V_n)_n2N is decreasing, we have that for alln2N

W =conv

n[1 k=1

(V_k\E_k)[ [1 k=n

(V_k\E_k)

!

✓conv

n[1 k=1

(V_k\E_k)[V_n

!

=W_n.

SinceFis a Cauchy filter on (E,⌧_ind), there existsB2Fsuch thatB B ✓W and soB B ✓W_n,8n2N. On the other hand we haveB\M_n6=;,8n2N, as both B and M_n belong toF. Hence, for all n2N we get

B (B\Mn)✓B B ✓Wn, which implies

B ✓W_n+ (B\M_n)✓W_n+M_n and so

B\E_n✓(W_n+M_n)\E_n^(1.14)= ;.

Therefore, we have got that B \En = ; for all n 2 N and so that B = ;, which is impossible asB 2F. Hence, (1.13) must hold true.

Example I: The space of polynomials

Letn2Nand x:= (x₁, . . . , x_n). Denote by R[x] the space of polynomials in the n variables x₁, . . . , x_n with real coefficients. A canonical algebraic basis forR[x] is given by all the monomials

x^↵:=x^↵₁¹· · ·x^↵_nⁿ, 8↵= (↵₁, . . . ,↵_n)2Nⁿ0.

For any d 2 N0, let Rd[x] be the linear subpace of R[x] spanned by all monomials x^↵ with |↵|:=Pn

i=1↵_i d, i.e.

Rd[x] :={f 2R[x]|degf d}.

Since there are exactly ^n+d_d monomials x^↵ with|↵|d, we have that dim(Rd[x]) = (d+n)!

d!n! ,

and so that R^d[x] is a finite dimensional vector space. Hence, by Tychono↵

Theorem (see Corollary 3.1.4 in TVS-I) there is a unique topology ⌧_e^d that

makesRd[x] into a Hausdor↵t.v.s. which is also complete and so Fr´echet (as it topologically isomorphic toR^dim(Rd[x])equipped with the euclidean topology).

As R[x] := S₁

d=0Rd[x], we can then endow it with the inductive topol- ogy ⌧_ind w.r.t. the family of F-spaces (Rd[x],⌧_e^d) :d2N0 ; thus (R[x],⌧_ind) is a LF-space and the following properties hold (proof as Sheet 3, Exercise 1):

a) ⌧_ind is the finest locally convex topology on R[x],

b) every linear map f from (R[x],⌧_ind) into any t.v.s. is continuous.

Example II: The space of test functions

Let ⌦✓R^d be open in the euclidean topology. For any integer 0s 1, we have defined in Section1.2the setC^s(⌦) of all real valueds times continuously di↵erentiable functions on ⌦, which is a real vector space w.r.t. pointwise addition and scalar multiplication. We have equipped this space with the C^s-topology (i.e. the topology of uniform convergence on compact sets of the functions and their derivatives up to orders) and showed that this turnsC^s(⌦) into a Fr´echet space.

Let K be a compact subset of ⌦, which means that it is bounded and closed inR^dand that its closure is contained in⌦. For any integer 0s 1, consider the subset Cc^k(K) ofC^s(⌦) consisting of all the functionsf 2C^s(⌦) whose support lies in K, i.e.

C^sc(K) :={f 2C^s(⌦) :supp(f)✓K},

wheresupp(f) denotes the support of the functionf on⌦, that is the closure in⌦ of the subset{x2⌦:f(x)6= 0}.

For any integer 0  s  1, Cc^s(K) is always a closed linear subspace of C^s(⌦). Indeed, for any f, g 2 C^sc(K) and any 2 R, we clearly have f+g2C^s(⌦) and f 2C^s(⌦) but alsosupp(f+g)✓supp(f)[supp(g)✓K and supp( f) = supp(f) ✓ K, which gives f +g, f 2 Cc^s(K). To show thatCc^s(K) is closed inC^s(⌦), it suffices to prove that it is sequentially closed as C^s(⌦) is a F-space. Consider a sequence (fj)j2N of functions in Cc^s(K) converging tof in theC^s topology. Then clearlyf 2C^s(⌦) and since all the f_j vanish in the open set ⌦\K, obviously their limit f must also vanish in

⌦\K. Thus, regarded as a subspace of C^s(⌦), Cc^s(K) is also complete (see Proposition 2.5.8 in TVS-I) and so it is itself an F-space.

Let us now denote byCc^s(⌦) the union of the subspacesCc^s(K) asK varies in all possible ways over the family of compact subsets of⌦, i.e. Cc^s(⌦) is linear subspace ofC^s(⌦) consisting of all the functions belonging toC^s(⌦) which have a compact support (this is what is actually encoded in the subscript c). In particular, the space Cc¹(⌦) (smooth functions with compact support in ⌦)

is called space of test functions and plays an essential role in the theory of distributions.

We will not endow Cc^s(⌦) with the subspace topology induced by C^s(⌦), but we will consider a finer one, which will turnCc^s(⌦) into an LF-space. Let us consider a sequence (K_j)_j2N of compact subsets of ⌦ s.t. K_j ✓K_j+1,8j 2N and S₁

j=1Kj = ⌦. (Sometimes is even more advantageous to choose the K_j’s to be relatively compact i.e. the closures of open subsets of⌦such that K_j ✓K˚_j+1,8j 2Nand S₁

j=1K_j =⌦.) Then Cc^s(⌦) = S₁

j=1Cc^s(Kj), as an arbitrary compact subset K of ⌦ is contained inK_jfor some sufficiently largej. Because of our way of defining the F-spaces Cc^s(K_j), we have that Cc^s(K_j) ✓ Cc^s(K_j+1) and Cc^s(K_j+1) induces on the subset Cc^s(Kj) the same topology as the one originally given on it, i.e. the subspace topology induced onCc^s(K_j) byC^s(⌦). Thus we can equipCc^s(⌦) with the inductive topology ⌧_ind w.r.t. the sequence of F-spaces {C^sc(K_j), j 2N}, which makesCc^s(⌦) an LF-space. It is easy to check that⌧_inddoes not depend on the choice of the sequence of compact setsK_j’s provided they fill⌦.

Note that (Cc^s(⌦),⌧_ind) is not metrizable (see Sheet 3, Exercise 2).

Proposition 1.3.10. For any integer 0  s  1, consider Cc^s(⌦) endowed with the LF-topology ⌧_ind described above. Then we have the following contin- uous injections:

Cc¹(⌦)!Cc^s(⌦)!Cc^{s 1}(⌦), 80< s <1.

Proof. Let us just prove the first inclusion i:Cc¹(⌦) ! Cc^s(⌦) as the others follows in the same way. As Cc¹(⌦) = S₁

j=1Cc¹(K_j) is the inductive limit of the sequence of F-spaces (Cc¹(K_j))_j2N, where (K_j)_j2N is a sequence of compact subsets of ⌦ such that K_j ✓ K_j+1,8j 2 N and S₁

j=1K_j = ⌦, by Proposition 1.3.5we know that i is continuous if and only if, for any j 2N, e_j := i Cc¹(K_j) is continuous. But from the definition we gave of the topology on each Cc^s(K_j) and Cc¹(K_j), it is clear that both the inclusions ij : Cc¹(Kj) ! Cc^s(Kj) and sj :Cc^s(Kj) ! Cc^s(⌦) are continuous. Hence, for each j2N,e_j =s_j i_j is indeed continuous.

1.4 Projective topologies and examples of projective limits

Let {(E_↵,⌧_↵) :↵2A} be a family of locally convex t.v.s. over the field Kof real or complex numbers (Ais an arbitrary index set). LetE be a vector space over the same fieldKand, for each↵2A, letf↵ :E !E↵be a linear mapping.

The projective topology ⌧_proj on E w.r.t. the family{(E_↵,⌧_↵, f_↵) :↵ 2A} is the coarsest topology on E for which all the mappings f_↵ (↵ 2 A) are continuous.

A basis of neighbourhoods of a point x2E is given by:

B^proj(x) :=

(\

↵2F

f_↵¹(U↵) : F ✓Afinite, U↵ nbhood off↵(x) in (E↵,⌧↵),8↵2F )

. Since the f_↵ are linear mappings and the (E_↵,⌧_↵) are locally convex t.v.s.,

⌧_proj on E has a basis of convex, balanced and absorbing neighbourhoods of the origin satisfying conditions (a) and (b) of Theorem 4.1.14 in TVS-I; hence (E,⌧_proj) is a locally convex t.v.s..

Proposition 1.4.1. Let E be a vector space over Kendowed with the projec- tive topology⌧_proj w.r.t. the family{(E_↵,⌧_↵, f_↵) :↵2A}, where each(E_↵,⌧_↵) is a locally convex t.v.s. over K and each f↵ a linear mapping from E to E↵. Then ⌧_proj is Hausdor↵ if and only if for each 0 6= x 2 E, there exists an

↵2Aand a neighbourhood U_↵ of the origin in(E_↵,⌧_↵)such that f_↵(x)2/ U_↵. Proof. Suppose that (E,⌧proj) is Hausdor↵ and let 0 6= x 2 E. By Propo- sition 2.2.3 in TVS-I, there exists a neighbourhood U of the origin in E not containing x. Then, by definition of ⌧_proj there exists a finite subset F ✓ A and, for any ↵ 2 F, there exists U↵ neighbourhood of the origin in (E↵,⌧↵) s.t. T

↵2F f_↵¹(U_↵)✓U. Hence, asx /2U, there exists↵2F s.t. x /2f_↵¹(U_↵) i.e. f_↵(x) 2/ U_↵. Conversely, suppose that there exists an ↵2A and a neigh- bourhood of the origin in (E↵,⌧↵) such that f↵(x)2/ U↵. Thenx /2f_↵¹(U↵), which implies by Proposition 2.2.3 in TVS-I that⌧_projis a Hausdor↵topology, asf_↵¹(U_↵) is a neighbourhood of the origin in (E,⌧_proj) not containingx.

Proposition 1.4.2. Let E be a vector space over K endowed with the pro- jective topology ⌧proj w.r.t. the family {(E↵,⌧↵, f↵) : ↵ 2 A}, where each (E_↵,⌧_↵) is a locally convex t.v.s. over K and each f_↵ a linear mapping from E toE_↵. Let(F,⌧)be an arbitrary t.v.s. and u a linear mapping fromF into E. The mapping u : F ! E is continuous if and only if, for each ↵ 2 A, f_↵ u:F !E_↵ is continuous.

Proof. (Sheet 3, Exercise 3)

Example I: The product of locally convex t.v.s

Let{(E_↵,⌧_↵) :↵2A}be a family of locally convex t.v.s. The product topol- ogy ⌧_prod on E = Q

↵2AE_↵ (see Definition 1.1.18 in TVS-I) is the coarsest topology for which all the canonical projections p↵ : E ! E↵ (defined by p_↵(x) :=x_↵ for anyx= (x ) _2A2E) are continuous. Hence,⌧_prod coincides with the projective topology on E w.r.t.{(E_↵,⌧_↵, p_↵) :↵2A}.

Let us consider now the case when we have a total order on the index set A, {(E_↵,⌧_↵) :↵ 2A} is a family of locally convex t.v.s. over K and for any ↵  we have a continuous linear mapping g_↵ :E ! E↵. Let E be the subspace of Q

↵2AE_↵ whose elements x = (x_↵)_↵2A satisfy the relation x_↵ = g_↵ (x ) whenever a ↵  . For any ↵ 2 A, let f_↵ be the canonical projection p↵ :Q

↵2AE↵ ! E↵ restricted to E. The space E endowed with the projective topology w.r.t. the family {(E_↵,⌧_↵, f_↵) : ↵ 2A} is said to be the projective limit of the family {(E_↵,⌧_↵) : ↵ 2A} w.r.t. the mappings {g_↵ :↵, 2A,↵ }. If eachf↵(E) is dense inE↵ then the projective limit is said to be reduced.

Remark 1.4.3. There are even more general constructions of projective limits of a family of locally convex t.v.s. (even when the index set is not ordered) but in the following we will focus on a particular kind of reduced projective limits. Namely, given an index set A, and a family {(E_↵,⌧_↵) : ↵ 2 A} of locally convex t.v.s. over K which is directed by topological embeddings (i.e.

for any ↵, 2 A there exists 2 A s.t. E ⇢ E↵ and E ⇢ E ) and such that the set E := T

↵2AE_↵ is dense in each E_↵, we will consider the reduced projective limit (E,⌧_proj). Here, ⌧_proj is the projective topology w.r.t. the family {(E↵,⌧↵, i↵) :↵2A}, where eachi↵ is the embedding ofE into E↵.

Example II: The space of test functions

Let ⌦✓R^d be open in the euclidean topology. The space of test functions Cc¹(⌦), i.e. the space of all the functions belonging to C¹(⌦) which have a compact support, can be constructed as reduced projective limit of the kind introduced in Remark 1.4.3.

Consider the index set

T :={t:= (t₁, t₂) : t₁2N0, t₂2C¹(⌦) witht₂(x) 1, 8x2⌦} and for each t2T, let us introduce the following norm onCc¹(⌦):

k'kt:= sup

x2⌦

0

@t₂(x) X

|↵|t1

|(D^↵')(x)| 1 A.

For each t 2T, let Dt(⌦) be the completion of Cc¹(⌦) w.r.t. k·kt. Then as sets

Cc¹(⌦) = \

t2T

Dt(⌦).

Consider on the space of test functions Cc¹(⌦) the projective topology ⌧_proj w.r.t. the family {(Dt(⌦),⌧_t, i_t) : t 2 T}, where (for each t 2 T) ⌧_t denotes the topology induced by the normk·ktanditdenotes the natural embedding of Cc¹(⌦) into Dt(⌦). Then (Cc¹(⌦),⌧_proj) is the reduced projective limit of the family {(Dt(⌦),⌧_t, i_t) :t2T}.

Using Sobolev embeddings theorems, it can be showed that the space of test functions Cc¹(⌦) can be actually written as projective limit of a family of weighted Sobolev spaces which are Hilbert spaces (see Chapter I, Section 3.10 of the book [Y. M. Berezansky, Selfadjoint Operators in Spaces of Functions of Infinite Many Variables, vol. 63, Trans. Amer. Math. Soc., 1986]).