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Math Help - Norm on C^1[a,b]

  1. #1
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    Norm on C^1[a,b]

    Let C^{1} [a,b] be the space of differentiable real valued functions f:[a,b] \to \mathbb {R} such that the derivative f':[a,b] \to \mathbb {R} is continuous. Given  f \in C^{1} [a,b] define ||f||_{C^{1}} by

    ||f||_{C^{1}}=||f||_{\infty} + ||f'||_{\infty}.

    (a) Prove that ||-||_{C^{1}} is a norm on C^{1} [a,b].

    Ok, so I have absolutely no idea how to do this. What defines a norm on a space of functions? How does one show that something is a norm?

    (b) Prove that C^{1} [a,b] is complete in this norm.

    Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.

    So let f_{n} in  C^{1} be a sequence of functions and suppose that  f_{n} is Cauchy.

    So for every \varepsilon > 0 there is an N such that n, l \geq N \implies ||f_{n} - f_{l}|| < \varepsilon.

    Need to show that f_{n} is convergent.

    Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?
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  2. #2
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    Quote Originally Posted by nmatthies1 View Post
    Let C^{1} [a,b] be the space of differentiable real valued functions f:[a,b] \to \mathbb {R} such that the derivative f':[a,b] \to \mathbb {R} is continuous. Given  f \in C^{1} [a,b] define ||f||_{C^{1}} by

    ||f||_{C^{1}}=||f||_{\infty} + ||f'||_{\infty}.

    (a) Prove that ||-||_{C^{1}} is a norm on C^{1} [a,b].

    Ok, so I have absolutely no idea how to do this. What defines a norm on a space of functions? How does one show that something is a norm?
    Do you know the definition of a norm? Norms are defined on vector spaces and C^1 is a vector space (by pointwise addition) over the reals.
    (b) Prove that C^{1} [a,b] is complete in this norm.

    Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.

    So let f_{n} in  C^{1} be a sequence of functions and suppose that  f_{n} is Cauchy.

    So for every \varepsilon > 0 there is an N such that n, l \geq N \implies ||f_{n} - f_{l}|| < \varepsilon.

    Need to show that f_{n} is convergent.

    Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?
    A norm is what determines if a space is complete. Completeness is a property of a metric, every norm induces a metric given by d(x,y)=||x-y||. Now if you change a metric (or a norm), the space may not be complete. Take the metric d(x,y)=|\frac{1}{x}-\frac{1}{y}| on [1,\infty) (which is complete under the Euclidian metric). Under this metric the sequence of natural numbers is Cauchy, but it coverges to 0 which is not in the set.

    As for the solution, what can you say about ||f_n-f_m||_\infty and ||f'_n-f'_m||_\infty. What happens to Cauchy sequences in (C^1[0,1],||\cdot||_\infty)?
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