TY - JOUR

T1 - Embedding into free topological vector spaces on compact metrizable spaces

AU - Gabriyelyan, Saak S.

AU - Morris, Sidney A.

N1 - Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by I, G, Q and Sk the closed unit interval, the Cantor space, the Hilbert cube Q=IN and the k-dimensional unit sphere for k∈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(Sk) can be embedded in V(Ik); (4) V(K) can be embedded in V(I) implies that K is finite-dimensional and metrizable.

AB - For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by I, G, Q and Sk the closed unit interval, the Cantor space, the Hilbert cube Q=IN and the k-dimensional unit sphere for k∈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(Sk) can be embedded in V(Ik); (4) V(K) can be embedded in V(I) implies that K is finite-dimensional and metrizable.

KW - Cantor space

KW - Compact

KW - Embedding

KW - Finite-dimensional

KW - Free locally convex space

KW - Free topological vector space

KW - Hilbert cube

KW - Zero-dimensional

UR - http://www.scopus.com/inward/record.url?scp=85033562340&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2017.09.008

DO - 10.1016/j.topol.2017.09.008

M3 - Article

AN - SCOPUS:85033562340

VL - 233

SP - 33

EP - 43

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -