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Given a vector space \(V\) over a field \(k\), a subset \(S\subseteq V\) is called 'linearly independent' if the only finite linear combination of elements of \(S\) to equal \(0\) is the trivial one, i.e. \(\forall T\subseteq S: |T|<\infty\Rightarrow (\forall \left[f:T\to k,\:v\mapsto a_v\right]:\sum_{v\in T}a_vv=0\Rightarrow \forall v\in T:a_v=0)\) - for any finite subset \(T\) of \(S\) and assignment of coefficients in \(k\) to each element of \(T\), if the sum of these vectors with these coefficients is the zero vector then all coefficients must have been 0. In the definition of a Schauder basis, the finiteness condition on \(T\) is dropped.