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

Table of Contents

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}

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.

On 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:

(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}

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