A sequence consisting of the [ilmath]n^\text{th} [/ilmath] terms of the sequences in a Cauchy sequence of elements in any [ilmath]\ell^p[/ilmath] space is itself a Cauchy sequence of complex numbers

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This theorem is a precursor to the little-L spaces are complete


Let [ilmath]p\in[1,+\infty]\subseteq\overline{\mathbb{R} } [/ilmath] be given and consider the [ilmath]\ell^p[/ilmath] space accordingly. So elements in [ilmath]\ell^p[/ilmath] are [ilmath](x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C} [/ilmath] such that certain properties hold.

Let [ilmath]\big((x_n^k)_{n\in\mathbb{N} }\big)_{k\in\mathbb{N} } \subseteq\ell^p[/ilmath] be a Cauchy sequence

  • i.e. [ilmath]\big((x^1_n)_{n\in\mathbb{N} },(x^2_n)_{n\in\mathbb{N} },\ldots,(x^k_n)_{n\in\mathbb{N} },\ldots\big)\subseteq\ell^p[/ilmath] is a Cauchy sequence

Then we claim[1]:

  • for all [ilmath]k\in\mathbb{N} [/ilmath] that [ilmath]x:\eq(x^n_k)_{n\in\mathbb{N} }\subseteq\mathbb{C} [/ilmath] is itself a Cauchy sequence
    • Notice that [ilmath]x:\eq(x^n_k)_{n\in\mathbb{N} } :\eq (x^1_k,x^2_k,\ldots,x^n_k,\ldots)[/ilmath] - the [ilmath]k^\text{th} [/ilmath] term of each element in [ilmath]x[/ilmath]


Grade: E
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  1. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha