Seminormed vector space

A seminorm induces a normed quotient

Let be a Seminormed vector space and let

Then the Quotient vector space carries a unique norm such that #m/def/anal/vec

Proof

Note that for any , . Now

proving absolute homogeneity. Similarly

proving the triangle inequality. Finally implies , so , thus we have positive definiteness.

#state/tidy | #lang/en | #SemBr