# 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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