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Space over R or C, for every real number p ≥ 1, the pnorm is indeed a norm The proof uses the following facts If q ≥ 1isgivenby 1 p 1 q =1, then (1) For all α,β ∈ R,ifα,β ≥ 0, then αβ ≤ αp p βq q (∗) (2) For any two vectors u,v ∈ E,wehave n i=1 u i v i≤u p v q (∗∗)T h u s m atrix A 3 o p eratin g upon th e v ecto r, X , y ield s an o th er v ecto r, X!ã /0 \ T he zero v ecto r, 0 = (0 1, is tran sfo rm ed into the zero v ecto r ag ain In g en eral,, th e d irectio n and m ag n itu d e of v ecto r, X 9 are d ifferen t fro m th o se o f v ecto r X!, (S ee "B eg in n ers1 C18 US Code § 922 Unlawful acts for any person, other than a licensed importer, licensed manufacturer, licensed dealer, or licensed collector, who does not reside in any State to receive any firearms unless such receipt is for lawful sporting purposes any firearm or armorpiercing ammunition to any person unless the licensee notes in his 2 A b c d e f g h i k l m n o p q r s t v x y z