K. Jänich, Topology. Chapter II: Topological Vector Spaces. 1. The Notion of a Topological Vector Space

Definition 1: Topological Vector Space Let \(K\) be either \(\mathbb{R}\) or \(\mathbb{C}\). A topological vector space \(V\) over \(K\) is a vector space equipped with a topology such that:

  • subtraction is a continuous function \(V^2 \rightarrow V\)
  • multiplication is a continuous function \(K \times V \rightarrow V\)

Definition 2: Topological Group A topological group is a group \(G\) equipped with a topology such that the function that maps \((x,y)\) to \(xy^{-1}\) is a continuous function \(G^2 \rightarrow G\).

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