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Topological Vector Space**

A topological vector space \((V,\mathcal{T},+,\cdot)\) over a topological field \(k\) is a \(k\)-vector space \((V,+,\cdot)\) such that \((V,\mathcal{T})\) is a topological space, and the maps \(+:V\times V\to V\) and \(\cdot:k\times V\to V\) are both continuous when \(V\times V\) and \(k\times V\) are endowed with the product topology.