Monomial basis for C^ω

The monomial basis for the vector space of analytic functions is given by

$${\displaystyle \{x^{n}\mid n\in \mathbb {N} \}.}$$

This basis is used in Taylor series, amongst others.

Monomial basis for polynomials

The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as ${\displaystyle a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}$ for some ${\displaystyle n\in \mathbb {N} }$, which is a linear combination of monomials.

Fourier basis for L²[0,1]

Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection

$${\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\mid n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\mid n\in \mathbb {N} \}\cup \{1\}}$$

forms a basis for L²[0,1].