61.11. EXISTENCE OF ABSTRACT WIENER SPACES 2047

Corollary 61.11.4 Let E be any real separable Banach space. Then there exists a se-quence, {ek} ⊆ E such that for any {ξ k} a sequence of independent random variables suchthat L (ξ k) = N (0,1), it follows

X (ω)≡∞

∑k=1

ξ k (ω)ek

converges a.e. and its law is a Gaussian measure defined on B (E). Furthermore, ||ek||E ≤λ k where ∑k λ k < ∞.

Proof: From the proof of Theorem 61.11.3 a basis for H is {λ kek} . Therefore, byTheorem 61.10.1, if {ξ k} is a sequence of independent N (0,1) random variables, then∑

∞k=1 ξ k (ω)λ kek converges a.e. to a random variable whose law is Gaussian. Also from

the proof of Theorem 61.10.1, each ek in that proof has the property that ||ek|| ≤ 1 becauseif ||ek||> 1, then you could consider β ≡(0,0, · · · ,1) and from the construction of the ek,you would need 1ek ∈ B(0,1) which is a contradiction. Thus ||λ kek|| ≤ λ k and changingthe notation, replacing λ kek with ek, this proves the corollary.