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Math Help - Normed linear Space

  1. #1
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    Normed linear Space

    I really need help for this . don't know where i need to start

    Question is
    Prove that C[a,b] is a normed linear space with ||f||= sup|f(x)| x is an element of closed interval [a,b] .


    #Here C[a,b] denotes the set of all continuous real valued functions defined on the closed interval [a,b].
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  2. #2
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    Quote Originally Posted by neset44 View Post
    I really need help for this . don't know where i need to start

    Question is
    Prove that C[a,b] is a normed linear space with ||f||= sup|f(x)| x is an element of closed interval [a,b] .


    #Here C[a,b] denotes the set of all continuous real valued functions defined on the closed interval [a,b].
    The problem, then, is just to show that all parts of the definitions are satisfied.

    "Linear Space". Show that both addition and scalar multiplication, f+ g and rf, where f and g are in C[a, b] and r is a number. Show that this has all the properties of a vectorspace:
    Addition is associative: (f+ g)+ h= f+ (g+ h)
    Addition is commutative: f+ g= g+ f
    scalar multiplication distributes over addition: r(f+ g)= rf+ rg
    r(sf)= (rs)f
    There exist a "zero" function so that f+ 0= f for all f in C[a, b]
    Every function has a "negative": for each f, there exist g such that f+ g= 0.

    Norm: ||f||= sup |f(x)|.

    Show that ||f|| is always non-negative and equal to 0 if and only if f is the "zero" function.
    Show that ||rf||= |r| ||f|| where f is a function in C[a,b] and r is a number.
    Show that ||f+ g||\le ||f||+ ||g||.
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