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For each positive integer $N$, let $x^{(N)}$ have its first $N$ coordinates equal to one and all later coordinates zero. This is an element of the l-p sequence space for every finite $p$, andNo constant can bound $\Vert x{\Vert }_p$ by a fixed multiple of $\Vert x{\Vert }_{\infty }$ for all $N$. Thus $\Vert \cdot {\Vert }_p$ and $\Vert \cdot {\Vert }_{\infty }$ are not Lipschitz equivalent.

For $x=(x_n)\in {\ell }^p$ with $1\le p<\infty$, letThenbecause the defining nonnegative series converges. Thus $F=c_{00}$ is dense in the l-p sequence space for every finite $p$; this is the density of finitely supported sequences in l-p.

It is not dense in the l-infinity sequence space. For the constant sequence $1=(1,1,\dots )$ and any finitely supported $y$, some tail of $y$ is zero, soHence $1$ does not belong to the closure of $F$.

The l-p sequence space ${\ell }^1$ has the Schur property. If a weakly null sequence stayed bounded below in norm, coordinatewise convergence and summability would select disjoint blocks containing almost all of successive terms. A sequence in ${\ell }^{\infty }$ matching their signs on those blocks would pair uniformly positively with a subsequence, contradicting weak convergence through the duality of l1 and l infinity.