Is \(\ell^{\infty} (\mathbb{R})\) A Banach Space?

Calculus Level 4

Let \(V\) be a vector space over \(\mathbb{R}\). A norm on \(V\) is a function \(\|\cdot \| : V \to \mathbb{R}\) satisfying the properties:

  • The norm is nonnegative: \(\|v\| \ge 0\) for all \(v\in V\), with equality if and only if \(v=0\).

  • The norm scales with vectors: \(\|cv\| = |c| \cdot \|v\|\) for all \(v\in V\) and \(c\in \mathbb{R}\).

  • The norm satisfies the triangle inequality: \(\|v+w\| \le \|v\| + \|w\|\) for all \(v, w\in V\).

A vector space equipped with a norm is called, unsurprisingly, a normed vector space.

Suppose \(V\) is a normed vector space. For \(v,w \in V\), define a function \(d: V\times V \to \mathbb{R}\) by \[d(v,w) = \|v-w\|.\] One can verify that this function \(d\) is a metric on \(V\), giving \(V\) the structure of a metric space.

If the metric space structure on \(V\) induced by the norm is complete (i.e., Cauchy sequences converge), then \(V\) is called a Banach space.

Consider the space \(\ell^{\infty} (\mathbb{R})\), consisting of bounded sequences of real numbers. This is a vector space over \(\mathbb{R}\), using coordinate-wise addition and scalar multiplication. One may define a norm on \(\ell^{\infty} (\mathbb{R})\) by setting \[\|(a_1, a_2, a_3, \cdots)\| := \sup_{i\ge 1} |a_i|.\] Under this norm, is \(\ell^{\infty} (\mathbb{R})\) a Banach space?


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