Continuous linear representation of topological group over topological field

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Definition

Suppose G is a topological group, K is a topological field, and V is a topological vector space over K. A continuous linear representation of G over V (and hence over K) is a linear representation \alpha: G \to GL(V), such that, when viewed as a map G \times V \to V, it is jointly continuous. In other words, it is continuous from G \times V (endowed with the product topology) to V.