# Thread: pointwise and uniforme convergence

1. ## pointwise and uniforme convergence

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

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

ii) $
f_k = kxe^{ - kx}
$

thanks!

2. Originally Posted by mms
Study pointwise and uniform convergence in the norm $
\left\| . \right\|_1
$
for the following sequences of functions in $
C\left( {\left[ {0,1} \right],\mathbb{R}} \right)
$

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

ii) $
f_k = kxe^{ - kx}
$

thanks!
What have you tried?

3. Originally Posted by mms
Study pointwise and uniform convergence in the norm $
\left\| . \right\|_1
$
for the following sequences of functions in $
C\left( {\left[ {0,1} \right],\mathbb{R}} \right)
$
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.