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Section 2.2 Topological Vector Spaces

Definition 2.2.1.
Let \(V\) be an abelian group and \(\bK\) a field. We say that \(V\) is a vector space over \(\bK\) if there is a ring homomorphism \(\psi\) of \(\bK\) into the endomorphisms of \(V\text{.}\)
The homomorphism \(\psi(\lambda)\) is called "multiplication by the scalar \(\lambda\)" and we will consistenly use the notation
\begin{equation*} \lambda\cdot v := (\psi(\lambda))(v). \end{equation*}
In this book, \(\bK\) will always be either \(\bR\) or \(\bC\text{.}\)
Definition 2.2.2.
A topological vector space is a vector space \(V\) with a topology \(\cU\) compatible with the linear structure, namely such that the sum of vectors and the multiplication by a scalar are continuous.
The indiscrete topology makes every vector space into a topological vector space.
Each euclidean space \(\bR^n\) is a topological vector space with respect to the standard topology of \(\bR^n\text{.}\)