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Math Help - Functional Analysis: Banach Spaces

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    Functional Analysis: Banach Spaces

    Show that C(\bar{\Omega}) with norm ||v||_{C(\bar{\Omega})}=\max\{|v(x)| : x\in\bar{\Omega}\} is a Banach space but that C^1(\bar{\Omega}) with the same norm is NOT a Banach space.

    So for the former I need to show that every Cauchy sequence in C(\bar{\Omega}) converges in said norm. So do I just define a random Cauchy sequence in the space, say \{v_n\}, so we have that ||v_m-v_n||\rightarrow 0 as m,n\rightarrow\infty. So now do I just substitute this directly into the norm given and show it must converge in that norm?

    Then the latter, how would I go about constructing a Cauchy sequence and then showing it doesn't converge in said norm? (Am I even on the right track?)
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    Quote Originally Posted by mathematicalbagpiper View Post
    Show that C(\bar{\Omega}) with norm ||v||_{C(\bar{\Omega})}=\max\{|v(x)| : x\in\bar{\Omega}\} is a Banach space but that C^1(\bar{\Omega}) with the same norm is NOT a Banach space.

    So for the former I need to show that every Cauchy sequence in C(\bar{\Omega}) converges in said norm. So do I just define a random Cauchy sequence in the space, say \{v_n\}, so we have that ||v_m-v_n||\rightarrow 0 as m,n\rightarrow\infty. So now do I just substitute this directly into the norm given and show it must converge in that norm?
    Yes, that is how to start. You have a Cauchy sequence of functions v_n, and you want to show that the sequence converges to a limit function. The first step is to construct the limit function, the second step is to show that the sequence converges to that limit.

    To construct the limit function, notice that for each x\in\bar{\Omega} the sequence of scalars v_n(x) is Cauchy and therefore converges because the scalar field is complete. Call this limit v(x). That defines your limit function v. You now have to show that \|v_n-v\|_{C(\bar{\Omega})}\to0.

    Quote Originally Posted by mathematicalbagpiper View Post
    Then the latter, how would I go about constructing a Cauchy sequence and then showing it doesn't converge in said norm? (Am I even on the right track?)
    For this, you need to think of a sequence of functions that converges in the C(\bar{\Omega})-norm to a non-differentiable function. You haven't said what the space \bar{\Omega} is. If it was something like the interval [1,1] of the real line, you might look for a sequence of polynomials that converges to |x|, for example.
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