Embedding into free topological vector spaces on compact metrizable spaces

作者：Gabriyelyan Saak S^*; Morris Sidney A

来源：Topology and Its Applications, 2018, 233: 33-43.

DOI：10.1016/j.topol.2017.09.008

• Free topological vector space
• Free locally convex space
• Embedding
• Finite-dimensional
• Zero-dimensional
• Compact
• Cantor space
• Hilbert cube

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.

• 出版日期2018-1-1