Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

The Theorem Levy Steinitz. Notation.
E is a real locally convex Hausdorff space.
Examples: R n , ℓp, 1 ≤ p ≤ ∞, Lp, 1 ≤ p ≤ ∞ (Banach spaces),
H(Ω), C ∞ (Ω) (Fréchet spaces: metrizable and complete), D, D ′ ,
H(K), A(Ω),...(more complicated spaces).
� uk is a convergent series and S( � uk) is its set of sums (of all
its convergent rearrangements).
Set of summing functionals
Γ( � uk) := {x ′ ∈ E ′ |
∞�
|〈x ′ , uk〉| < ∞} ⊂ E ′ .
The annihilator of G ⊂ E ′ is G ⊥ := {x ∈ E | 〈x, g〉 = 0 ∀g ∈ G}.
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José Bonet Rearrangement of series. The theorem of Levy-Steiniz.