I am an academic statistician working on a problem that involves some optimization in infinite dimensional Banach spaces. My specific question is as follows. We know that in Banach space $C[0,1]$ a weak convergence of the sequence of functions is equivalent to :1) pointwise convergence of this sequence and 2) uniform boundedness of the sup norm for the entire sequence. Are there any similar results/characterizations for sequences of functions from Banach spaces of $k$ times differentiable functions $C^{k}[0,1]$? I am especially interested in the case $k=1$ but a general answer would be even more exciting. I suspect that it would involve something like a uniform boundedness of a sequence of derivative suprema but I am not sure. Any comments/answers are welcome.
Michael