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Math Help - Isometric

  1. #1
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    Isometric

    Show that C[0,1] and C[a,b] are isometric.
    Solution:
    A mapping T of X into X' is said to be isometric if for all x, y in X
    d'(Tx,Ty)=d(x,y)
    C is continous [0,1] and [a,b] closed interval..
    The distance on Function space C[a,b] is d(x,y)=max|x(t)-y(t)|

    I couldnt procced from here....i need some help....

    thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    Show that C[0,1] and C[a,b] are isometric.
    Solution:
    A mapping T of X into X' is said to be isometric if for all x, y in X
    d'(Tx,Ty)=d(x,y)
    C is continous [0,1] and [a,b] closed interval..
    The distance on Function space C[a,b] is d(x,y)=max|x(t)-y(t)|

    I couldnt procced from here....i need some help....

    thanks
    What if you did a mapping that bijectively took [a,b]\to[0,1] in an increasing way, call this g, and you considered T:C[a,b]\to C[0,1] defined by Tf(x)=f(g(x))?
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  3. #3
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    Thank u very much...
    But how can we prove that the map is bijective...i mean how can we see that it is one-to-one and onto...
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    Thank u very much...
    But how can we prove that the map is bijective...i mean how can we see that it is one-to-one and onto...
    Have you tried to prove it?
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  5. #5
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    yes i thought about it..but i couldnt find a way...please can u help me...
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