Let || || : Cmxn R be a norm, defined for all m, n e...

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Let || · || : Cmxn R be a norm, defined for all m, n e N. Show that there always exists a c> 0 such that the constant multiple || |le := c|| • || defines a submultiplicative norm, i.e., ||AB ||e < || A||e|| B ||. for any A E Cmxn and B E C"XP (even if || || does not have this property). Find the constant c for the Hölder 0o-norm. Let || · || : Cmxn R be a norm, defined for all m, n e N. Show that there always exists a c> 0 such that the constant multiple || |le := c|| • || defines a submultiplicative norm, i.e., ||AB ||e < || A||e|| B ||. for any A E Cmxn and B E C"XP (even if || || does not have this property). Find the constant c for the Hölder 0o-norm.