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A normed vector space ${(X, \| \|_X)}$ automatically generates a topology, known as the norm topology or strong topology on ${X}$, generated by the open balls ${B(x,r) := \{ y \in X: \|y-x\|_X < r \}}$. A sequence ${x_n}$ in such a space converges strongly (or converges in norm) to a limit ${x}$ if and only if ${\|x_n-x\|_X \rightarrow 0}$ as ${n \rightarrow \infty}$. This is the topology we have implicitly been using in our previous discussion of normed vector spaces.

However, in some cases it is useful to work in topologies on vector spaces that are weaker than a norm topology. One reason for this is that many important modes of convergence, such as pointwise convergence, convergence in measure, smooth convergence, or convergence on compact subsets, are not captured by a norm topology, and so it is useful to have a more general theory of topological vector spaces that contains these modes. Another reason (of particular importance in PDE) is that the norm topology on infinite-dimensional spaces is so strong that very few sets are compact or pre-compact in these topologies, making it difficult to apply compactness methods in these topologies. Instead, one often first works in a weaker topology, in which compactness is easier to establish, and then somehow upgrades any weakly convergent sequences obtained via compactness to stronger modes of convergence (or alternatively, one abandons strong convergence and exploits the weak convergence directly). Two basic weak topologies for this purpose are the weak topology on a normed vector space ${X}$, and the weak* topology on a dual vector space ${X^*}$. Compactness in the latter topology is usually obtained from the Banach-Alaoglu theorem (and its sequential counterpart), which will be a quick consequence of the Tychonoff theorem (and its sequential counterpart) from the previous lecture.

The strong and weak topologies on normed vector spaces also have analogues for the space ${B(X \rightarrow Y)}$ of bounded linear operators from ${X}$ to ${Y}$, thus supplementing the operator norm topology on that space with two weaker topologies, which (somewhat confusingly) are named the strong operator topology and the weak operator topology.