weak convergence
Weak convergence is a gentler notion of getting close: instead of demanding that vectors approach each other in length, it only asks that every continuous linear measurement of them agree in the limit. Two vectors can be far apart in norm yet weakly near, provided no single functional can tell them apart in the limit. It is convergence as seen through every possible shadow rather than head-on.
Precisely, in a normed space X a sequence x_n converges weakly to x if f(x_n) -> f(x) for every bounded linear functional f in the dual X*. Norm convergence (||x_n - x|| -> 0, also called strong convergence) always implies weak convergence, but not conversely. The point of weak convergence is that it is far easier to achieve: bounded sequences in a reflexive space (for instance any Hilbert space, or L^p with 1 < p < infinity) always have weakly convergent subsequences, even though they may have no norm-convergent ones.
The classic warning is that the norm is only lower semicontinuous, not continuous, under weak limits: one has ||x|| <= liminf ||x_n||, and the inequality can be strict — mass can “escape” in the limit. In an infinite-dimensional Hilbert space the orthonormal sequence e_n converges weakly to 0 while keeping ||e_n|| = 1 for all n, a vivid demonstration that weak limits lose information that norm limits would preserve.
In l^2 the basis vectors e_n satisfy <e_n, y> = y_n -> 0 for every fixed y (since sum |y_n|^2 < infinity forces y_n -> 0), so e_n converges weakly to 0. Yet ||e_n - 0|| = 1 for all n, so e_n does NOT converge in norm. This single sequence separates the two notions cleanly.
The orthonormal sequence in l^2: weakly null but of constant norm 1, never norm-convergent.
In finite dimensions weak and strong convergence coincide, because there are only finitely many coordinates to test. The two split apart only in infinite dimensions, which is exactly where weak convergence earns its keep: it restores a form of compactness (bounded sequences have convergent subsequences) that the norm topology loses.