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Thread: Norm on C^1[a,b]

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

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

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

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

    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 $\displaystyle C^{1} [a,b]$ 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 $\displaystyle f_{n}$ in $\displaystyle C^{1} $ be a sequence of functions and suppose that $\displaystyle f_{n} $ is Cauchy.

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

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

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

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

    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 $\displaystyle C^{1} [a,b]$ 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 $\displaystyle f_{n}$ in $\displaystyle C^{1} $ be a sequence of functions and suppose that $\displaystyle f_{n} $ is Cauchy.

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

    Need to show that $\displaystyle f_{n}$ 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 $\displaystyle d(x,y)=||x-y||$. Now if you change a metric (or a norm), the space may not be complete. Take the metric $\displaystyle d(x,y)=|\frac{1}{x}-\frac{1}{y}|$ on $\displaystyle [1,\infty)$ (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 $\displaystyle ||f_n-f_m||_\infty$ and $\displaystyle ||f'_n-f'_m||_\infty$. What happens to Cauchy sequences in $\displaystyle (C^1[0,1],||\cdot||_\infty)$?
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