l-p vector space

$${\ell }^p=\{a=\{a_j{\}}_{j=1}^{\mathrm{\infty }}|||a|{|}_p<\mathrm{\infty }\}\$\$isthespaceof(infinite)sequences.Wedefinethe\${\ell }^p\$normas\$\$||a|{|}_p:=\{\begin{array}{l}(\sum _{i=1}^{\mathrm{\infty }}|a_i{|}^p{)}^{1/p}1\le p<\mathrm{\infty }\\ \underset{1\le j<\mathrm{\infty }}{sup}|a_j|p=\mathrm{\infty }\end{array}$$

${\left\{\frac{1}{j}\right\}}_{j=1}^{\mathrm{\infty }}\in {\ell }^p\mathrm{\forall }p>1$ but not for $p=1$