# Subspaces of the free topological vector space on the unit interval Academic Article

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### abstract

• For a Tychonoff space \$X\$, let \$\mathbb{V}(X)\$ be the free topological vector space over \$X\$, \$A(X)\$ the free abelian topological group over \$X\$ and \$\mathbb{I}\$ the unit interval with its usual topology. It is proved here that if \$X\$ is a subspace of \$\mathbb{I}\$, then the following are equivalent: \$\mathbb{V}(X)\$ can be embedded in \$\mathbb{V}(\mathbb{I})\$ as a topological vector subspace; \$A(X)\$ can be embedded in \$A(\mathbb{I})\$ as a topological subgroup; \$X\$ is locally compact.

• 2018