61.11. EXISTENCE OF ABSTRACT WIENER SPACES 2045

this being well defined because the {ek} are independent. Let the norm on H0 be denotedby |·|H0

. Let H1 be the completion of H0 with respect to this norm.I want to show that |·|H0

is stronger than ||·||. Suppose then that∣∣∣∣∣ n

∑k=1

β kek

∣∣∣∣∣H0

≤ 1.

It follows then from the definition of |·|H0that∣∣∣∣∣ n

∑k=1

β kek

∣∣∣∣∣2

H0

=n

∑k=1

β2k ≤ 1

and so from the construction of the ek, it follows that∣∣∣∣∣∣∣∣∣∣ n

∑k=1

β kek

∣∣∣∣∣∣∣∣∣∣< 1

Stated more simply, this has just shown that if h∈H0 then since∣∣∣h/ |h|H0

∣∣∣H0≤ 1, it follows

that||h||/ |h|H0

< 1

and so||h||< |h|H0

.

It follows that the completion of H0 must lie in E because this shows that every Cauchysequence in H0 is a Cauchy sequence in E. Thus H1 embedds continuously into E and isdense in E. Denote its norm by |·|H1

.Now consider the nuclear operator,

A =∞

∑k=1

λ kek⊗ ek

where each λ k > 0 and ∑k λ k < ∞. This operator is clearly one to one. Also it is clear theoperator is Hilbert Schmidt because ∑k λ

2k < ∞. Let

H ≡ AH1.

and for x ∈ H, define|x|H ≡

∣∣A−1x∣∣H1

.

Since each ek is in H it follows that H is dense in E. Note also that H ⊆H1 because A mapsH1 to H1.

Ax≡∞

∑k=1

λ k (x,ek)ek