This is the first of three lectures on the topic of weak convergence we are showing from our 'Functional Analysis' 3rd year course. The lectures build upon results on Banach spaces and their duals that are covered earlier in the course. In this lecture Melanie explains why familiar arguments from finite dimensional analysis break down in infinite dimensional spaces since bounded sequences cannot be expected to have convergent subsequences, so she introduces an alternative weaker notion of convergence. You can watch many other student lectures via our main Student Lectures playlist (also check out specific student lectures playlists):    • Oxford Mathematics Student Lectures -...   All first and second year lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.