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Math Help - pointwise and uniforme convergence

  1. #1
    mms
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    pointwise and uniforme convergence

    Study pointwise and uniform convergence in the norm <br />
\left\| . \right\|_1 <br />
for the following sequences of functions in <br />
C\left( {\left[ {0,1} \right],\mathbb{R}} \right)<br />

    i) <br />
f_k = \left\{ \begin{gathered}<br />
kx\,\,\,\,if\,x \in \left[ {0,1/k} \right] \hfill \\<br />
\left( {kx} \right)^{ - 1} \,\,if\,x \in \left[ {1/k,1} \right] \hfill \\ <br />
\end{gathered} \right.<br />

    ii) <br />
f_k = kxe^{ - kx} <br />

    thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mms View Post
    Study pointwise and uniform convergence in the norm <br />
\left\| . \right\|_1 <br />
for the following sequences of functions in <br />
C\left( {\left[ {0,1} \right],\mathbb{R}} \right)<br />

    i) <br />
f_k = \left\{ \begin{gathered}<br />
kx\,\,\,\,if\,x \in \left[ {0,1/k} \right] \hfill \\<br />
\left( {kx} \right)^{ - 1} \,\,if\,x \in \left[ {1/k,1} \right] \hfill \\ <br />
\end{gathered} \right.<br />

    ii) <br />
f_k = kxe^{ - kx} <br />

    thanks!
    What have you tried?
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  3. #3
    Super Member
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    México
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    Quote Originally Posted by mms View Post
    Study pointwise and uniform convergence in the norm <br />
\left\| . \right\|_1 <br />
for the following sequences of functions in <br />
C\left( {\left[ {0,1} \right],\mathbb{R}} \right)<br />
    Wait, what does this even mean? Pointwise and uniform convergence are independent of the norm you give to your functional space (they only depend on the metric of the domain and codomain). And what is \| \cdot \| _1 ? Is it \| x\| _1 = \int_{0}^{1} x(t)dt ? Please give all the information.
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