Weighted PLB-spaces of continuous functions arising as tensor products of a Fréchet and a DF-space (Q542061)

The author deals with weighted PLB-spaces of continuous functions on a locally compact, \(\sigma\)-compact topological space. These are the spaces of the form \(\text{proj}_N\text{ind}_nC(a_{N,n})_0(X)\), where \(C(a_{N,n})_0(X)\) is a weighted Banach space of continuous functions vanishing at \(\infty\) with the supremum norm and \(A=(a_{N,n})\) is a double sequence of weights which is increasing in \(N\) and decreasing in \(n\). The author is interested in the situation when \(X=X_1\times X_2\) is a product of two spaces and \(A^i\) \((i=1,2)\) are increasing sequences of weights on \(X_i\). Then he defines a double sequence \(A\) on \(X\) by the formula \(a_{n,N}(x_1,x_2):=\frac{a^1_N(x_1)}{a^2_n(x_2)}\). Theorem 3 shows when such a defined PLB-space \((AC)_0(X)\) is ultrabornological. Here the \((DN)-(\Omega)\) type conditions of Vogt play an important role. Further it is shown (Proposition 2) when \((AC)_0(X)\) is an injective tensor product of a Fréchet space and a DF-space. The author also proves in Theorem 4 when the above injective tensor product is ultrabornological. Again, conditions of type \((DN)\) and \((\Omega)\) appear.