Let Is the sequence convergent in ?
First I tried to check if this sequence is Cauchy and I calculated integral of the sequence over . I got ). Then I'm stuck can anyone help?
If the -limit of the sequence exists, then it must be equal to the pointwise limit, which is the zero function. So all you need to do is to check whether as . In fact, if you make the substitution , you see that , and so the sequence does not converge in the -norm.
What about the same , is is convergent in C[0,1] with sup norm? I think it converges to .
I agree, the pointwise limit is . To show that this is the limit in the sup norm, you need to show that uniformly on [0,1]. So look at the difference and see if that can be made small for all , provided that is large enough.