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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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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