$\ell^\infty$ and $\ell^1$

Show that $\ell^\infty$ and $\ell^1$ are normed linear spaces.

Solution:

Since $\ell^p$ is the collection of real sequences $a=(a_1,a_2, ... )$ for which $\sum_{k=1}^{\infty} |a_k|^p < \infty$ and in $\ell^\infty$, $\sup_{1 \leq k < \infty} |a_k|$. Does this merely require checking the properties of a norm (Triangle-Inequality, Positive Homogeneity, Non-negativity)?

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excuse the mistake in the title. – Jake Casey Jan 26 '13 at 16:56
you can always edit your answer, as soon as you discover mistakes, see the edit option below your question. – clark Jan 26 '13 at 17:00
You may also have to verify that they are linear spaces. – David Mitra Jan 26 '13 at 17:02
Or, show that the space $F$ of all infinite sequences is a linear space (hopefully you have this result in hand), and that $\ell_1$ and $\ell_\infty$ are subspaces of $F$ (non-empty and closed under addition and scalar multiplication). – David Mitra Jan 26 '13 at 17:17
to be precis you need to verify that they are linear spaces, scalar multiplication and addition, and that the map to $\mathbb{R}$ is indeed a norm with the properties you just stated. – user25470 Jan 26 '13 at 17:18

Recall first that the set of all real sequences $\mathbb{R}^{\mathbb{N}^*}=\{a=(a_n)_{n\geq 1}\}\;|\;a_n\in\mathbb{R}\}$ is a real vector space.

Now denote $\|a\|_\infty:=\sup_{n\geq 1}|a_n|$ and $\|a\|_p:=\left(\sum_{n\geq 1}|a_n|^p\right)^{1/p}$ for $p\geq 1$. For every $a\in\mathbb{R}^{\mathbb{N}^*}$, it is clear that both $\|a\|_\infty$ and $\|a\|_p$ are nonnegative or infinite.

Next define $\ell_\infty:=\{a\in \mathbb{R}^{\mathbb{N}^*}\;|\;\|a\|_\infty<\infty\}$ and $\ell_p:=\{a\in \mathbb{R}^{\mathbb{N}^*}\;|\;\|a\|_p<\infty\}$.

Then note that the null sequence $(0,0,0,\ldots)$ belongs to both $\ell_\infty$ and $\ell_p$, so that they are nonempty subsets of $\mathbb{R}^{\mathbb{N}^*}$.

Now, indeed, it only remains to prove that $\|a\|_\infty$ and $\|a\|_p$ satisfy the three axioms of a norm http://en.wikipedia.org/wiki/Norm_(mathematics):

1)positive homogeneity

2)triangle inequality

3)separation.

It will follow from 1) and 2) that both $\ell_\infty$ and $\ell_p$ are stable under linear combinations, hence vector (linear) subspaces of $\mathbb{R}^{\mathbb{N}^*}$. This will show that they are normed vector (linear) spaces.

1) is easy for both $\ell_\infty$ and $\ell_p$.

2) is easy for $\ell_\infty$ and for $\ell_p$, it follows from the Minkowski inequality: http://en.wikipedia.org/wiki/Minkowski_inequality

3) is easy for both $\ell_\infty$ and $\ell_p$.

I hope this helps.

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