Editing Normed vector space (section)

==Normed spaces as quotient spaces of seminormed spaces==

The definition of many normed spaces (in particular, [[Banach space]]s) involves a seminorm defined on a vector space and then the normed space is defined as the [[Quotient space (linear algebra)|quotient space]] by the subspace of elements of seminorm zero.  For instance, with the [[Lp space|<math>L^p</math> spaces]], the function defined by
<math display=block>\|f\|_p = \left( \int |f(x)|^p \;dx \right)^{1/p}</math>
is a seminorm on the vector space of all functions on which the [[Lebesgue integral]] on the right hand side is defined and finite.  However, the seminorm is equal to zero for any function [[Support (mathematics)|supported]] on a set of [[Lebesgue measure]] zero.  These functions form a subspace which we "quotient out", making them equivalent to the zero function.