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It is proved that every a-compact locally convex metric linear space containing a topological copy of the Hilbert cube Q is homeomorphic to E the linear span of the Hilbert cube Q, in the Hilbert space 12. Let us fix the following representation of the Hilbert cube Q in the Hilbert space 12: Q = {(X,) E 12: IX,I < n-2 for every n}. Consider the space E = span(Q) C 12. We note that E is a a-compact linear dense subspace of 12. The purpose of this note is to prove the following fact: THEOREM. Every a-compact locally convex metric linear space E containing a topological copy of the Hilbert cube Q is homeomorphic to E. Moreover, if E is the completion of E, then the pairs (E, E) and (12, E) are homeomorphic. The Theorem resolves the outstanding problem LS3 in [4] posed by Anderson in connection with the fact that the assertion of the Theorem was known to be true under the additional assumption that the copy of the Hilbert cube above was a convex set (see: Anderson [1], Bessaga-Pelczyn'ski [2] and Toruniczyk [5]). Let us recall that the classical theorem of Keller-Klee [3, p. 1001 states that every metrizable infinite-dimensional compact convex subset K of a locally convex space is homeomorphic to Q. We complete the paper with an example of a a-compact dense linear subspace of 12 which contains a topological copy of Q and does not contain any infinite-dimensional compact convex subset. This shows that the Theorem cannot be reduced to the above-mentioned result of [1, 2, 5]. The proof of the Theorem involves the technique of absorbing sets. We will use the approach of West [8] and Torunczyk [6] instead of the original approaches of Anderson (cap-set technique [1]) and Bessaga-Pelczynski (skeleton technique [2]). According to [8, 6] a a-compact set M is an absorbing set (for compacta) in a space X homeomorphic to 12 if the following condition is satisfied: (abs) every map f: Q -+ X such that f IB is an embedding of a closed set B C Q into M C X can be approximated by embeddings g: Q -+ M with gIB = fIB. The space E may serve as an example of an absorbing set in 12 (see [3, p. 275]). We intend to show that the space E is also an absorbing set in the completion E homeomorphic, by the Kadec-Anderson theorem [3, p. 189], to 12. Then the assertion of our Theorem will follow from the theorem on equivalence of absorbing sets [8, 6] stating that: if Mi is an absorbing set in Xi (i = 1, 2), then the pairs (X1, M1) and (X2, M2) are homeomorphic. Received by the editors July 6, 1981. 1980 Mathematics Subject Clasification. Primary 57N17, 57N20.

Sigma-compact locally convex metric linear spaces universal for compacta are homeomorphic