Basis

In finite dimensional spaces
A collection of vectors $$\{\mathbf{e}_k\}_{k=1}^N$$ (Def. 1.3.2 ) is a basis for a (complex) vector space $$V$$ if $$\text{span}\{\mathbf{e}_k\}_{k=1}^N = V$$ and the vectors $$\{\mathbf{e}_k\}_{k=1}^N$$ are linearly independent.

Basis in normed vector space (possibly infinite dimensional)
A collection of vectors $$\{\mathbf{v}_k\}_{k=1}^N$$ (Def. 2.5.4 ) is a basis for a normed vector space $$V$$ if for each $$\mathbf{v} \in V$$ there exists unique scalar coefficients $$\{c_k\}_{k=1}^\infty$$ such that

$$\mathbf{v} = \sum_{k=1}^{\infty} c_k \mathbf{v}_k$$. This definition is also the Schauder basis.