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

Calculus Level 3

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

  • The norm is nonnegative: v0\|v\| \ge 0 for all vVv\in V, with equality if and only if v=0v=0.
  • The norm scales with vectors: cv=cv\|cv\| = |c| \cdot \|v\| for all vVv\in V and cRc\in \mathbb{R}.
  • The norm satisfies the triangle inequality: v+wv+w\|v+w\| \le \|v\| + \|w\| for all v,wVv, w\in V.

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

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

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

Consider the space (R)\ell^{\infty} (\mathbb{R}), consisting of bounded sequences of real numbers. This is a vector space over R\mathbb{R}, using coordinate-wise addition and scalar multiplication. One may define a norm on (R)\ell^{\infty} (\mathbb{R}) by setting (a1,a2,a3,)=supi1ai.\|(a_1, a_2, a_3, \cdots)\| = \sup_{i\ge 1} |a_i|. Under this norm, is (R)\ell^{\infty} (\mathbb{R}) a Banach space?


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