ℓ1, ℓp, ℓ∞, and c0 Sequences Normed Linear Spaces Review

# ℓ1, ℓp, ℓ∞, and c0 Sequences Normed Linear Spaces Review

We will now review some of the recent material regarding the $\ell^1$, $\ell^p$, $\ell^{\infty}$, and $c_0$ sequence spaces.

• On The ℓ1 Sequences Normed Linear Space page we defined the $\ell^1$ sequence space to be the set of all sequences whose series is absolutely convergent, and we defined the $1$-norm on $\ell^1$ by:
(1)
\begin{align} \quad \| (a_i) \|_1 = \sum_{i=1}^{\infty} |a_i| \end{align}
• We proved that $(\ell^1, \| \cdot \|_1)$ is a normed linear space.
• On The ℓp Sequences Normed Linear Space page we defined the $\ell^p$ sequence space to be the set of all sequences whose series is $p$-absolutely convergent, and we defined the $p$-norm on $\ell^p$ by:
(2)
\begin{align} \quad \| (a_i) \|_p = \left ( \sum_{i=1}^{\infty} |a_i|^p \right )^{1/p} \end{align}
• We noted that $(\ell^p, \| \cdot \|_p)$ is a normed linear space. The only this that we did not prove is the triangle inequality for the $p$-norm, which is proven by Minkowski's inequality for $\ell^p$ later on.
(3)
\begin{align} \quad \| (a_i) \|_{\infty} = \sup_{i \geq 1} \{ |a_i| \} \end{align}
• We proved that $(\ell^{\infty}, \| \cdot \|_{\infty})$ is a normed linear space.
• Lastly, on The c0 Sequences Normed Linear Space page we defined the $c_0$ sequence space to be the set of all sequences converging to $0$ and we defined the $\infty$ norm on $c_0$ (similarly as above) by:
(4)
\begin{align} \quad \| (a_i) \|_{\infty} = \sup_{i \geq 1} \{ |a_i| \} \end{align}
• We proved that [[$(c_0, \| \cdot \|_{\infty})$] is a normed linear space.