Embedding into free topological vector spaces on compact metrizable spaces

For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by II, G, Q and S-k the closed unit interval, the Cantor space, the Hilbert cube Q = I-N and the k-dimensional unit sphere for k is an element of N, respectively. The main result is that V(R) can be embedded as a topological vector space in V(I). It is also shown that for a compact Hausdorff space K: (1) V(K) can be embedded in V(G) if and only if K is zero-dimensional and metrizable; (2) V(K) can be embedded in V(Q) if and only if K is metrizable; (3) V(S-k) can be embedded in V(I-k); (4) V(K) can be embedded in V(I) implies that K is finite-dimensional and metrizable. (C) 2017 Elsevier B.V. All rights reserved.