little-lp Space

Last updated Nov 1, 2022

Let $1 \leq p \leq \infty$. We define $l_{p}$ for $p < \infty$ as $$l_{p} := {({x}{n}){n=1}^{\infty} \subset \mathbb{R} : \sum\limits_{n=1}^{\infty} |x_{n}|^{p} < \infty}$$ equipped with the norm $||x||{p} := \left(\sum\limits{n=1}^{\infty}x^{p}\right)^{\frac{1}{p}}$ for $x \in l_{p}$. If $p = \infty$, then we define $l_\infty$ as $$l_{\infty} := {({x}{n}){n=1}^{\infty} \subset \mathbb{R} : \sup\limits_{n \in \mathbb{N}} |x_{n}| < \infty}$$ equipped with the norm $||x||{\infty} := \sup\limits{n \in \mathbb{N}} |x_{n}|$ for $x \in l_{\infty}$.

1. little-lp is Banach

# TODO

• Redo this by setting up
  • Extended Norm
  • little-lp Norm
  • Supremum Norm
  • Finite Normed Vectors form a Normed Vector Space

TODO as an Lp Space over the Counting Measure.