ℓ1, ℓ∞, and ℓp Normed Linear Spaces

ℓ1, ℓ∞, and ℓp Normed Linear Spaces

Definition: The $\ell^1$-Linear Space is defined to be the set $\displaystyle{\ell^1 = \left \{ (a_i)_{i=1}^{\infty} : \sum_{i=1}^{\infty} |a_i| < \infty \right \}}$ with the norm $\| \cdot \|_1 : \ell_1 \to [0, \infty)$ defined for all $(a_i)_{i=1}^{\infty} \in \ell^1$ by $\displaystyle{\| (a_i)_{i=1}^{\infty} \|_1 = \sum_{i=1}^{\infty} |a_i|}$.

So $\ell^1$ consists of all sequences of real (or complex) numbers whose sum is absolutely convergent, and the norm of the sequence is defined to be the absolutely convergent sum.

Definition: The $\ell^{\infty}$-Linear Space is defined to be the set $\displaystyle{\ell^{\infty} = \left \{ (a_i)_{i=1}^{\infty} : (a_i)_{i=1}^{\infty} \: \mathrm{is \: bounded} \right \}}$ with the norm $\| \cdot \|_{\infty} : \ell^{\infty} \to [0, \infty)$ defined for all $(a_i)_{i=1}^{\infty} \in \ell^{\infty}$ by $\displaystyle{\| (a_i)_{i=1}^{\infty} \|_{\infty} = \sup_{i \geq 1} \{ |a_i| \}}$.

So $\ell^{\infty}$ consists of all sequences of real (or complex) numbers that are bounded, and the norm of the sequence is defined to be the supremum of the absolute value of the terms in the sum.

Definition: For $1 < p < \infty$, the $\ell^p$-Linear Space is defined to be the set $\displaystyle{\ell^p = \left \{ (a_i)_{i=1}^{\infty} : \sum_{i=1}^{\infty} |a_i|^p < \infty \right \}}$ with the norm $\| \cdot \|_p : \ell^p \to [0, \infty)$ defined for all $(a_i)_{i=1}^{\infty} \in \ell^p$ by $\displaystyle{\| (a_i)_{i=1}^{\infty} \|_p = \left ( \sum_{i=1}^{\infty} |a_i|^p \right )^{1/p}}$.

So $\ell^p$ consists of all sequences of real (or complex) numbers whose sum of is $p$-absolutely convergent, and the norm of the sequence is defined to be the $p^{\mathrm{th}}$ roots of the $p$-absolutely convergent sum.

While we have made the following definitions above, it still needs to be verified that $\ell^1$, $\ell^{\infty}$, and $\ell^p$ are all normed linear spaces.

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