= Normed vector space
{title2=$(X,\|\cdot\|)$}
{wiki}
= Normed space
{synonym}
A normed vector space is a <vector space> equipped with a <norm>, whose induced <metric> is $d(x,y)=\lVert x-y\rVert$. The <norm> must be positive on nonzero <vectors>, scale by the absolute value of a <scalar>, and satisfy the <triangle inequality>. These conditions make distance compatible with addition and scaling.
On $C([0,1])$, both the <supremum norm> $\|f\|_\infty=\max_{0\leq x\leq1}|f(x)|$ and the <L2 norm> $\|f\|_2=(\int_0^1|f(x)|^2\,dx)^{1/2}$ are genuine <norms>. A nonzero <continuous function> stays nonzero on an interval of positive length, so its squared <integral> cannot vanish. For $f(x)=x$, they give $1$ and $1/\sqrt3$, respectively. Always $\|f\|_2\leq\|f\|_\infty$, but the shrinking peaks below show that no uniform reverse bound exists. Unlike the <finite-dimensional equivalence of norms>, infinite-dimensional <vector spaces> can support inequivalent <norm topologies>.
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