The Weak Topology

Abstract

When we say that a sequence f _n in the space C[0, 1] converges to f, we mean that ‖f _n − f‖ → 0 as n → ∞; and this is the same as saying f _n converges to f uniformly. There are other notions of convergence that are weaker, and still very useful in analysis. This is the motivation for studying different topologies on spaces of functions, and on general Banach spaces.