Notes:Ell^p(C) is complete for p between one and positive infinity inclusive

Statement

[ilmath]\forall p\in[1,+\infty]\subseteq\overline{\mathbb{R} } [/ilmath] the space [ilmath]\ell^p(\mathbb{C})[/ilmath] is complete.

Proof

Leammas

Lemma 1:

• Let [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} }\subseteq\ell^p(\mathbb{C})[/ilmath] be given
  • Suppose [ilmath](\mathbf{x}_n)_{n\in\mathbb{N} } [/ilmath] is a Cauchy sequence
    • Write them out as follows:

      [ilmath]\begin{array}{ccccc} \mathbf{x}_1 & \eq & (x_1^m)_{m\in\mathbb{N} } & \eq & x_1^1,\ x_1^2,\ \ldots ,\ x_1^k,\ \ldots \\ \mathbf{x}_2 & \eq & (x_2^m)_{m\in\mathbb{N} } & \eq & x_2^1,\ x_2^2,\ \ldots,\ x_2^k,\ \ldots \\ \vdots \end{array}[/ilmath]

    • Now consider the [ilmath]k^\text{th} [/ilmath] column of this table, this gives us the sequence: [ilmath](x_m^k)_{m\in\mathbb{N} } [/ilmath]
      • We claim that [ilmath](x_m^k)_{m\in\mathbb{N} } [/ilmath] is a Cauchy sequence itself

Lemma 2:

Let [ilmath](X,d)[/ilmath] be a metric space, then [ilmath]A\in\mathcal{P}(X)[/ilmath] is bounded in [ilmath]X[/ilmath] if and only if:

• [ilmath]\forall x\in X\exists C\in\mathbb{R}_{>0}\forall a\in A[d(a,x)\le C][/ilmath]