A Schwartz space E is universal if every Schwartz space is topologically isomorphic to a linear subspace of some power E' of E. In this paper concrete examples of universal Schwartz spaces are exhibited.

In this paper we show (see Theorem 2.4) that a separated locally convex (topological vector) space is a Schwartz space if and only if it is topologically isomorphic to a compact projective limit (see 2.1 below) of co-spaces, where co 1-respectively, :~] denotes the Banach space of zero-convergent 1-respectively, bounded] scalar-valued sequences equipped with the supremum norm II • l[. The proof is based upon the representation theorems for precompact linear operators given in [8].1. The co-Extension Property 1.1. A linear operator T • E -, F from one separated locally convex space into another is precompact [respectively, compact] if T transforms a neighborhood of 0 in E into a precompact [respectively, relatively compact] subset of F. Following 1-8, 2.17 and 2.9] we say that a metrizable locally convex space F has the co-extenion property if each precompact linear operator from a linear subspace of c o into F can be extended to a continuous linear operator from co into F. Following 1-3, p. 337] we say that a linear operators T : E~F from one separated locally convex space into another admits a compact factorization through co, if there exist compact linear operator P:E--*c o and Q:co-+F such that T = Q P . Following [8, 2.4] we say that a sequence {Yn} in a separated locally convex space F is strongly summable if, for each ~ in :o~, the series Y-~,Yn converges in F and {2;~,yn : 11 ~ II oo -< 1, ~ in :®} is bounded in E The proof of our structure theorem (2.4) will be based on the following consequence of [8, Theorem 2.18]. 1.2. Proposition. For a metrizable locally convex space F the following are equivalent: (a) F has the c:extension property. (b) Each precompact linear operator from a separated locally convex space E into F admits a compact factorization through co. (c) For each precompact linear operator T : E -* F from a separated locally convex space E into F, there is a sequence 2 in c,, an equicontinuous sequence {an} in the topological dual E' of E and a strongly sutureable sequence {Yn} in F such that for each x in E T x = ~,;tn(x, an~yn.

It is shown that every compact operator T:E-*F between Banach spaces admits a compact factorization (T=QP where P:E-+c and Q:c-*Fare compact) through a closed subspace c of the Banach space c0 of zero-convergent sequences.

It is shown that every compact operator T:E-*F between Banach spaces admits a compact factorization (T=QP where P:E-+c and Q:c-*Fare compact) through a closed subspace c of the Banach space c0 of zero-convergent sequences.

It is shown that c0 (the Banach space of zeroconvergent sequences) is the only Banach space with basis that satisfies the following property: For every compact operator T:c"-->-E from c0 into a Banach space E, there is a sequence X in c0 and an unconditionally summable sequence {yn} in E such that Tn=J An/i"j" for each /i in c0. This result is then used to show that a linear operator T:E-*F from a locally convex space E into a Fréchet space Fhas a representation of the form Tx= y A"", where A is a sequence in c", {o"} is an equicontinuous sequence in the topological dual E' of E and {y,} is an unconditionally summable sequence in F, if and only if Tcan be "compactly factored" through c0.