## Abstract

For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by I, 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∈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.

Original language | American English |
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Pages (from-to) | 33-43 |

Number of pages | 11 |

Journal | Topology and its Applications |

Volume | 233 |

DOIs | |

State | Published - 1 Jan 2018 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

- Geometry and Topology