Does changing length change convergence?

Source scope: §3.4. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Measure one vector three ways

For $x=(3,4)$, the one-norm is $7$, the two-norm is $5$, and the infinity-norm is $4$. Each measures length using a different unit ball.

A norm on a real vector space satisfies, for all $x,y$ and real $\alpha$,

A vector space with a norm is a normed space. Its induced metric is $d(x,y)=\|x-y\|$. Positivity gives separation; $\|x-y\|=\|-(y-x)\|=\|y-x\|$ gives symmetry; writing $x-z=(x-y)+(y-z)$ gives the triangle inequality.

A family of lengths

On $\mathbb R^n$,

The first two are special cases of the $p$-norm. To see the infinity limit, put $M=\max_i|x_i|$. Then $M\le\|x\|_p\le n^{1/p}M$, and $n^{1/p}\to1$.

For $0<p<1$, the formula is not a norm. With $u=(1,0)$ and $v=(0,1)$, $\|u+v\|_p=2^{1/p}>2=\|u\|_p+\|v\|_p$. This supplies the counterexample requested in the source.

Equivalent does not mean equal

On any fixed finite-dimensional real space, any two norms admit constants $\alpha,\beta>0$ such that

Consequently convergence in either norm implies convergence in the other: apply the upper bound to the differences from the limit, and use the lower bound for the reverse implication. The source states norm equivalence without a proof and warns that it can fail in infinite-dimensional function spaces.

Source footnote 4: A normed space complete in its induced metric is a Banach space. Complete means every Cauchy sequence has a limit in the space; being Cauchy means its terms eventually become arbitrarily close to one another.

Transfer

If a sequence tends to zero in the two-norm in $\mathbb R^{10}$, can it fail to tend to zero in the one-norm?