On the Fréchet space L p −

Mean ergodic operators and reflexive Fréchet lattices

“…[22]). We also mention`p + , 1 p < 1 (see [23]) and L p , 1 < p < 1 (see [11]). Further examples are L ) consisting of the p-th power m-integrable (respectively, weakly m-integrable) functions are Fréchet lattices (see [9], [10], [25]).…”

“…It should be observed that the centre Z (L p ) may be identi…ed with L 1 (0; 1), acting on L p via multiplication. Evidently, L p is not discrete and hence (as is well known; [11]), it is not a Montel space (that is, L p does not satisfy condition (ii) of Corollary 5.4). According to [1], Proposition 2.11, the space L p is not uniformly mean ergodic.…”

Some embedding theorems for Hörmander–Beurling spaces

“…Furthermore, since all complemented subspace of a quojection is a quojection (see [28]), H is a quojection (actually H ∞ r=1 X r where each X r coincides with some l p (L q j ([0, 1]))), E < B loc p,k (Ω, E) and E is not a quojection (see [3]), it follows that H does not contain any complemented copy of B loc p,k (Ω, E). 5.…”

“…We will use the Fréchet spaces l q + = p>q l p and L q − = p<q L p ([0, 1]) (these spaces have an interest in the structure theory of Fréchet spaces and are primary and have all nuclear Λ 1 (α)-spaces as complemented subspaces, see [27,3] …”

The Fatou Completion of a Fréchet Function Space and Applications

“…Indeed, L 0 (µ) is a σ -Dedekind complete Riesz [5] The . Consider a net 0 ≤ u λ ↑ that is topologically bounded in F. We need to show that sup λ u λ exists in F and that q n (sup λ u λ ) = sup λ q n (u λ ) for all n ∈ N. The proof proceeds as for normed spaces [23,…”