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:

\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.

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**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:

\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.

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**The ℓ∞ Sequences Normed Linear Space**page we defined the $\ell^{\infty}$ sequence space to be the set of all bounded sequences, and we defined the $\infty$-norm on $\ell^{\infty}$ by:

\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:

\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.