Why are the sequence spaces denoted by $\ell^p$?

Question

I have read functional analysis and metric spaces and conceptually, I have no problem at all with the sequence spaces. However, I do wonder why do sequence spaces are denoted by $\ell^p$?

Answer 1

As stated in the comments, the notation is in honor of H. Lebesgue. If $(X,\mathcal{A},\mu)$ is a measure space, then we can define the $p$-space of integrable functions as $$L^p(\mu)=\{f:X\to\mathbb{C}:\int_X|f|^pd\mu<\infty\}$$ Now there is a special measure that we can define on $\mathbb{N}$, called the counting measure: if $E\subset\mathbb{N}$, then $\mu(E)=$ how many elements $E$ has. It can be easily proved that $\mu$ is a measure and that $\ell^p=L^p(\mu)$, hence these spaces come from a large variety of spaces, called the $L^p$ spaces.