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Countable projective limits of countable inductive limits, so-calledPLB-spaces, of weighted Banach spaces of continuous functions have recentlybeen investigated by Agethen, Bierstedt and Bonet, who analyzed locally convexproperties in terms of the defining double sequence of weights. We complementtheir results by considering a defining sequence which is the product of twosingle sequences. By associating these two sequences with a weightedFr\'{e}chet, resp. LB-space of continuous functions or with two weightedFr\'{e}chet spaces (by taking the reciprocal of one of the sequences) we derivea representation of the PLB-space as the tensor product of a Fr\'{e}chet and aDF-space and exhibit a connection between the invariants (DN) and ($\Omega$)for Fr\'{e}chet spaces and locally convex properties of the PLB-space resp. ofthe forementioned tensor product.

Weighted PLB-spaces of continuous functions arising as tensor products of a Fr\'echet and a DF-space