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

Calculus Level 3

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

×