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Span**

Given a vector space \(V,+,\cdot\) over a field \(k\), the span of a subset \(M\subseteq V\), written \(\mathrm{span}_k(M)\) or \(\langle M\rangle\), is the set of all finite linear combinations of elements of \(M\), \(\mathrm{span}_k(M)=\left\{\sum_{r=1}^n a_rv_r\middle|n\in\mathbb{Z}_{\geq0},\:a_r\in k,\:v_r\in M\right\}\). The span is also the smallest (by inclusion) sub-vector space of \(V\) to contain \(M\) or, equivalently, the intersection of all subspaces which contain \(M\). If \(V\) is further a topological vector space, then the closure of the span, \(overline{\mathrm{span}_k(M)}\), is the set of all convergent (possibly infinite) linear combinations of elements of \(M\).