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ℱ-bases with brackets and with individual brackets in Banach spaces

Tomasz Kochanek (2012)

Studia Mathematica

We provide a partial answer to the question of Vladimir Kadets whether given an ℱ-basis of a Banach space X, with respect to some filter ℱ ⊂ 𝒫(ℕ), the coordinate functionals are continuous. The answer is positive if the character of ℱ is less than 𝔭. In this case every ℱ-basis is an M-basis with brackets which are determined by an element of ℱ.

ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

Denny H. Leung, Wee-Kee Tang (2006)

Studia Mathematica

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X = T [ ( θ ₙ , ₙ ) n = 1 ∞ ] : (1) Every block subspace of X contains an ℓ ¹ - ω -spreading model, (2) The Bourgain ℓ¹-index I b ( Y ) = I ( Y ) > ω ω for any block subspace Y of X, (3) l i m ₘ l i m s u p ₙ θ m + n / θ ₙ > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...

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