of S(V)k can be identified with a polynomial of degree k in the commuting variables {xi}i∈Ω and with coefficients in K. Hence, S(V) is identified with the commutative polynomial ring K[xi:i∈Ω].
The universal property ofS(V) easily follows from the universal property of T(V).
Proposition 4.3.4. Let V be a vector space over K. For any unital commu-tative K−algebra(A,∗) and any linear mapψ:V →A, there exists a unique K−algebra homomorphism ψ¯ : S(V) → A such that the following diagram commutes
V A
S(V)
ψ
ψ¯
i.e. ψ¯V=ψ.
Corollary 4.3.5. Let V be a vector space over K. The algebraic dual V∗ of V is algebraically isomorphic to Hom(S(V),K).
Proof. For any α ∈ Hom(S(V),K) we clearly have α V∈ V∗. On the other hand, by Proposition 4.3.4, for any ` ∈ V∗ there exists a unique ¯` ∈ Hom(S(V),K) such that ¯`V=`.
4.4 An lmc topology on the symmetric algebra of a lc TVS
Let V be a vector space over K. In this section we are going to explain how a locally convex topology τ on V can be naturally extended to a locally convex topology τ on the symmetric algebra S(V) (see [14]). Let us start by considering the simplest possible case, i.e. when τ is generated by a single seminorm.
Suppose now that ρ is a seminorm on V. Starting from the seminorm ρ on V, we are going to construct a seminorm ¯ρ onS(V) in three steps:
1. For k ∈ N, let us consider the projective tensor seminorm on V⊗k see Theorem4.2.2, i.e.
ρ⊗k(g) := (ρ⊗ · · · ⊗ρ
| {z }
ktimes
)(g)
= inf ( N
X
i=1
ρ(gi1)· · ·ρk(gik) :g=
N
X
i=1
gi1⊗ · · · ⊗gik, gij ∈V, N ∈N )
.
2. Denote byπk:V⊗k→S(V)kthe quotient mapπ restricted toV⊗kand
Define ρ0 to be the usual absolute value on K.
3. For anyh∈S(V), sayh=h0+· · ·+h`,fk ∈S(V)k,k= 0, . . . , `, define
We refer toρ as theprojective extension of ρ toS(V).
Proposition 4.4.1. ρ is a seminorm on S(V) extending the seminorm ρ on V and ρ is also submultiplicative i.e. ρ(f·g)≤ρ(f)ρ(g), ∀f, g ∈S(V)
To prove this result we need an essential lemma:
Lemma 4.4.2. Let i, j ∈ N, f ∈ S(V)i and g ∈ S(V)j. If k = i+j then
It is quite straightforward to show that ρ is a seminorm onS(V). Indeed
4.4. An lmc topology on the symmetric algebra of a lc TVS ρ is submultiplicative. Let f =Pm
i=0fi,g=Pn
Let us now consider (S(V), ρ) and any other submultiplicative seminormed unital commutative K−algebra (A, σ). Ifα : (S(V), ρ)→(A, σ) is linear and continuous, then clearly α V: (V, ρ) → (A, σ) is also continuous. However, if ψ : (V, ρ) → (A, σ) is linear and continuous, then the unique extension ψ given by Proposition4.3.4need not be continuous . All one can say in general is the following lemma.
Lemma 4.4.3. If ψ : (V, ρ) → (A, σ) is linear and continuous, namely ∃ C > 0 such that σ(ψ(v)) ≤ Cρ(v) ∀ v ∈ V, then for any k ∈ N we have σ(ψ(g))≤Ckρk(g) ∀ g∈S(V)k.
Proof.
Let k ∈ N and g ∈ S(V)k. Suppose g = PN
i=1gi1· · ·gik with gij ∈ V for j= 1, . . . , N. Then ψ(g) =PN
i=1ψ(gi1)· · ·ψ(gik), and so σ(ψ(g))≤σ
N
X
i=1
ψ(gi1)· · ·ψ(gik)
!
≤
N
X
i=1
σ(ψ(fi1))· · ·σ(ψ(gik))
≤
N
X
i=1
Cρ(gi1)· · ·Cρ(gik) =Ck
N
X
i=1
ρ(gi1)· · ·ρ(gik).
As this holds for any representation of g, we getσ(ψ(g))≤Ckρk(g).
Proposition 4.4.4. If ψ : (V, ρ) → (A, σ) has operator norm ≤1, then the induced algebra homomorphism ψ : (S(V), ρ) → (A, σ) has operator norm
≤σ(1).
Recall that given a linear operator L between two seminormed spaces (W1, q1) and (W2, q2) we define the operator norm of Las follows:
kLk:= sup
w∈W1
q1(w)≤1
q2(L(w)).
Proof.
Supposeσ 6≡0 onA (if this is the case then there is nothing to prove). Then there exists a∈A such thatσ(a)>0. This together with the fact thatσ is a submultiplicative seminorm gives that
σ(1)≥1. (4.4)
Since kψk ≤ 1, we have that σ(ψ(v)) ≤ ρ(v), ∀v ∈ V. Then we can apply Lemma4.4.3 and get that
∀k∈N, g∈S(V)k, σ(ψ(g))≤ρk(g) (4.5)
4.4. An lmc topology on the symmetric algebra of a lc TVS
Using the properties we have showed for the projective extension ¯ρ of ρ to S(V), we can easily pass to the case when V is endowed with a locally convex topology τ (generated by more than one seminorm) and to study how to extend this topology to S(V) in a such a way that the latter becomes an lmc TA.
Letτ be any locally convex topology on a vector space V overKand let P be a directed family of seminorms generatingτ. Denote by τ the topology on S(V) determined by the family of seminormsQ:={nρ:ρ∈ P, n∈N}.
Proposition 4.4.5. τ is an lmc topology on S(V) extending τ and is the finest lmc topology on S(V) having this property.
Proof. By definition ofτ and by Proposition 4.4.1, it is clear thatQ is a di-rected family of submultiplicative seminorms and so thatτ is an lmc topology on S(V) extending τ.It remains to show that τ is the finest lmc topology with extending τ toS(V). Let µan lmc topology on S(V) s.t. µV=τ, i.e.
µ extends τ to S(V). Suppose that µ is finer than τ. Let S be a directed family of submultiplicative seminorms generating µand consider the identity map id : (V, τ) → (V, µ V). As by assumption µ V=τ, we have that id is continuous and so by Theorem 4.6.3-TVS-I (applied for directed families of seminorms) we get that:
∀s∈ S,∃n∈N,∃ρ∈ P : s(v) =s((id(v))≤nρ(v),∀v∈V.
Consider the embedding i : (V, nρ) → (S(V), q). Then kik ≤ 1 and so, by Proposition 4.4.4, the unique extension ¯i : (S(V), nρ) → (S(V), s) of i is continuous with k¯ik ≤q(1). This gives that
s(f)≤s(1)nρ(f),∀f ∈S(V).
Hence, all s∈ F are continuous w.r.t.τ and so µmust be coarser thanτ.
Chapter 5
Short overview on the moment problem
In this chapter we are going to consider always Radon measures on Hausdorff topological spaces, i.e. non-negative Borel measures which are locally finite and inner regular.
5.1 The classical finite-dimensional moment problem
Let µbe a Radon measure onR. We define the n−th moment of µ as mµn:=
Z
R
xnµ(dx)
If all moments ofµexist and are finite, then we can associate toµthe sequence of real numbers (mµn)n∈N0, which is said to be themoment sequence ofµ. The moment problem exactly addresses the inverse question:
Problem 5.1.1 (The one-dimensionalK−Moment Problem (KMP)).
Given a closed subset K of R and a sequence m= (mn)n∈N0 of real numbers, does there exist a Radon measureµonRs.t. for anyn∈N0we havemn=mµn
and µ is supported on K, i.e.
mn= Z
R
xnµ(dx)
| {z }
n-th moment ofµ
,∀n∈N0 and supp(µ)⊆K?
If such a measureµdoes exist we say thatµis aK−representing measure form or that m is represented byµon K.
Note that there is a bijective correspondence between the set RN0 of all sequences of real numbers and the set (R[x])∗ of all linear functional fromR[x]
toR.
RN0 → (R[x])∗
(mn)n∈N0 7→ Lm: R[x] → R
p(x) :=P
j
pjxj 7→ Lm(p) :=P
j
pjmj. (L(xn))n∈N0 ←[ L
In virtue of this correspondence, we can always reformulate the KMP in terms of linear functionals
Problem 5.1.2 (The one-dimensionalK−Moment Problem (KMP)).
Given a closed subset K of Rand a linear functional L:R[x]→R, does there exists a Radon measure µon R s.t.
L(p) = Z
R
p(x)µ(dx),∀p∈R[x] and supp(µ)⊆K?
As before, if such a measure exists we say that µ is a K−representing measure forL and that it is a solution to the K−moment problem forL.
Clearly one can generalize the one-dimensional KMP to higher dimension by consideringR[x] :=R[x1, . . . , xd] for somed∈N(see [15, Section 5.2.2]).
Problem 5.1.3 (Thed-dimensionalK−Moment Problem (KMP)).
Given a closed subset K of Rd and a linear functional L : R[x] → R, does there exists a Radon measureµ on Rd s.t.
L(p) = Z
Rd
p(x)µ(dx),∀p∈R[x] and supp(µ)⊆K? It is then very natural to ask the following:
Questions
• What if we have infinitely many variables, i.e. we considerR[xi:i∈Ω]
where Ω is an infinite index set?
• What if instead of real variables we consider variables in a generic R−vector spaceV (even infinite dimensional)?
• What if instead of the polynomial ringR[x] we take any unital commu-tativeR−algebraA?
All these possible generalization of the moment problem usually go under the name ofinfinite dimensional moment problem.