= Banach space
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A Banach space is a <normed vector space> over the <real numbers> or <complex numbers> that is <complete> for the <metric> $d(x,y)=\|x-y\|$. Explicitly, whenever a <sequence> $(x_n)$ satisfies
$$
\forall\varepsilon>0\;\exists N\;\forall m,n\geq N:\quad\|x_m-x_n\|<\varepsilon,
$$
there is an element $x$ of the same space with $\|x_n-x\|\to0$. The hypothesis says that the approximations eventually agree to any requested accuracy; <completeness> says that they describe an actual object in the space. The <Cauchy sequence> condition can be checked before anyone knows what that object is.
This makes a <Banach space> a natural setting for constructing solutions by successive improvements. If corrections $v_k$ have $\sum_k\|v_k\|<\infty$, their <partial sums> form a <Cauchy sequence> by the <triangle inequality>, and <completeness> supplies their sum. In fact, convergence of every such absolutely summable vector <series> characterizes a <Banach space>. The <contraction mapping theorem> uses precisely this mechanism: each correction is smaller than the preceding one by a fixed factor.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf[Richard Melrose, Functional Analysis, chapter 1, section 3, Proposition 1.2: completeness and summable series]{ref}
Every <normed vector space> of finite <dimension> is a <Banach space>. Infinite-dimensional examples include the <absolutely summable sequence space> and the <l-infinity sequence space>; the choice of <norm> remains part of the specification. A collection of increasingly accurate finite descriptions can require an infinite description for its limit, as the <finitely supported sequence> example below demonstrates. <Completion of a normed space> adds these missing limits while preserving all original distances.
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