Banach spaces

A normed vector space $$V$$ is called a Banach space (Def 3.1.4 ) if every Cauchy sequence in $$V$$ converges to some $$\mathbf{v}\in V$$.

Examples to Banach space are $$\mathbb{R}^n, \mathbb{C}^n$$. The space of continuous functions on a bounded interval, $$C[a,b]$$, is also a Banach space. Moreover, the spaces $$\ell^p(\mathbb{N})$$ ($$p \in [1,\infty)$$) defined as

$$\ell^p(\mathbb{N}) := \{ \{x_k\}_{k=1}^\infty : x_k \in \mathbb{C}, \sum_{k=1}^\infty |x_k|^p < \infty \}$$

are also Banach spaces w.r.t. the norm $$||\mathbf{x}||=\left(\sum_{k=1}^\infty |x_k|^p\right)^{1/p}$$