TITLE: Norm for pointwise convergence
QUESTION [36 upvotes]: Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?

REPLY [5 votes]: The first answer does not reply exactly to the question:
indeed, the topology of the pointwise convergence is not metrizable; however, there could be another (metrizable or not) topology with the same convergent sequences! This phenomen happens with the Schur property of $\ell^1$.
hardmath's example answers completely the question. Interestingly, the following article shows also that there is no metric having the same set of convergent sequences as the pointwise convergent sequences:
http://www.ams.org/journals/proc/1951-002-01/S0002-9939-1951-0040647-7/S0002-9939-1951-0040647-7.pdf