# Embeddings of free topological vector spaces Academic Article

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### abstract

• It is proved that the free topological vector space \$\mathbb{V}([0,1])\$ contains an isomorphic copy of the free topological vector space \$\mathbb{V}([0,1]^{n})\$ for every finite-dimensional cube \$[0,1]^{n}\$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval \$[0,1]\$ to general metrisable spaces. Indeed, we prove that the free topological vector space \$\mathbb{V}(X)\$ does not even have a vector subspace isomorphic as a topological vector space to \$\mathbb{V}(X\oplus X)\$, where \$X\$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

• 2019