Properties of norm on C[a, b]?

Let C[a, b] denote the set of all continuous, real-valued functions on the interval [a, b]. For f∈C[a, b], let ||f||∞ denote max a%26lt;=t=%26lt;b |f(t)|.





If f, g∈C[a, b] show that ||fg||∞ %26lt;= ||f||∞ ||g||∞ . Also show that ||max{f,g}||∞ %26lt;= max{||f||∞, ||g||∞ }, and ||f||∞ %26lt;=||g||∞ whenever |f| %26lt;= |g|.





Thanks a lot for your help.

Properties of norm on C[a, b]?
This is just a matter of definitions, since ||f||∞ is just the maximum value of |f| on [a, b].





a) ||fg||∞ = maximum value of |f*g| on [a,b]. But this is obviously going to have to be less than or equal to:


(maximum of |f| on [a,b]) *(maximum of |g| on [a,b])


because these maxima are evaluated separately: you can shop around "for the best deal" on the interval. But then that is exactly the same as:


||f||∞ * ||g||∞


So:


||fg||∞ =%26lt; ||f||∞ * ||g||∞





b) In exactly the same way,


||max{f,g}||∞ = maximum value of max(|f| , |g| ) on [a,b] =%26lt; max(maximum value of |f| on [a,b], maximum value of |g| on [a,b]) = max(||f||∞, ||g||∞).





c) Since:


|f|(x) =%26lt; |g|(x) ,


max of |f|(x) on [a,b] =%26lt; max of |g|(x) on [a,b] , so:


||f||∞ =%26lt; ||g||∞
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