1. Special classes of topological vector spaces in the subsequent ones, but a priori we do not know if E

in the subsequent ones, but a priori we do not know if E_n is isomorphically embedded in E, i.e. if the topology induced by ⌧_ind on E_n is identical to the topology ⌧n initially given on En. This is indeed true and it will be a consequence of the following lemma.

Lemma 1.3.3. Let X be a locally convex t.v.s., X₀ a linear subspace of X equipped with the subspace topology, and U a convex neighbourhood of the origin in X₀. Then there exists a convex neighbourhood V of the origin in X such that V \X₀=U.

Proof.

AsX₀ carries the subspace topology induced by X, there exists a neighbour- hoodW of the origin inXsuch thatU =W\X₀. SinceX is a locally convex t.v.s., there exists a convex neighbourhood W0 of the origin in X such that W₀✓W. LetV be the convex hull ofU[W₀. Then by construction we have that V is a convex neighbourhood of the origin in X and that U ✓V which implies U =U \X0 ✓V \X0. We claim that actually V \X0 =U. Indeed, let x 2 V \X₀; as x 2 V and as U and W₀ are both convex, we may write x=ty+ (1 t)z with y2U, z 2W₀ and t2[0,1]. If t= 1, then x=y 2U and we are done. If 0t <1, then z= (1 t) ¹(x ty) belongs toX₀ and so z 2 W₀ \X₀ ✓ W \X₀ = U. This implies, by the convexity of U, that x2U. Hence,V \X₀ ✓U.

Proposition 1.3.4.

Let (E,⌧_ind) be an LF-space with defining sequence {(E_n,⌧_n) :n2N}. Then

⌧ind En⌘⌧n,8n2N. Proof.

(✓) LetU be a neighbourhood of the origin in (E,⌧_ind). Then, by definition of ⌧_ind, there existsV convex, balanced and absorbing neighbourhood of the origin in (E,⌧_ind) s.t. V ✓U and, for eachn2N,V \E_n is a neighbourhood of the origin in (E_n,⌧_n). Hence, ⌧_ind E_n✓⌧_n,8n2N.

(◆) Givenn2N, let U_n be a convex, balanced, absorbing neighbourhood of the origin in (E_n,⌧_n). Since E_n is a linear subspace ofE_n+1, we can apply Lemma 1.3.3 (for X = E_n+1, X₀ = E_n and U = U_n) which ensures the existence of a convex neighbourhood U_n+1 of the origin in (E_n+1,⌧_n+1) such that Un+1\En =Un. Then, by induction, we get that for any k2 N there exists a convex neighbourhood U_n+k of the origin in (E_n+k,⌧_n+k) such that U_n+k\E_{n+k 1} =U_{n+k 1}. Hence, for anyk2N, we get U_n+k\E_n =U_n. If we consider now U := S₁

k=1U_n+k, then U \En = Un. Furthermore, U is a

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

E_n

!

= [

n2N

\

V2F(o)

(V \E_n) = [

n2N

\

Un2Fn(o)

U_n={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 E_n of u to E_n is continuous.

2. A linear form on E is continuous if and only if its restrictions to each En 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 results is instead typical of LF-spaces as it heavily relies on the completeness of the t.v.s. of the defining sequence.

Theorem 1.3.6. 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 byFE(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 toG. 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_p6=; (1.13) This property ensures that the family

Gp :={A\E_p : 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 E_p=⌧_p,Gp is a Cauchy filter on (E_p,⌧_p).

Hence, the completeness ofEpguarantees that there existsx2Ep s.t. Gp !x.

This implies that alsoG !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_n2 G s.t. A_n\E_n=;. By the definition ofG, this means that

8n2N,9Mn2F, Vn2FE(o),s.t. (Mn+Vn)\En=;. (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.

SinceF is a Cauchy filter on (E,⌧_ind), there existsB 2Fsuch 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 to F. Hence, for all n2Nwe 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 \E_n = ; 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, . . . , xn 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 |↵|:=P_n

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(R^d[x]) = (d+n)!

d!n! ,

and so that Rd[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 makesR^d[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 (R^d[x],⌧_e^d) :d2N⁰ ; thus (R[x],⌧_ind) is a LF-space and the following properties hold (proof as Exercise 1, Sheet 3):

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.