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I know for a banach space $\mathbf{X}$ that the norm $||\cdot||$ produces a topology for the space. I also know this is considered "strong". I also understand that the dual, $\mathbf{X}^\ast$, is the set of all continuous linear functionals. I also understand that the set of functions in $\mathbf{X}^\ast$ can be used to create a topology on $\mathbf{X}$ such that they are all continuous. This one being called "Weak topology", I don't understand this, why is it called weak? Why is it viewed as being "weaker" than the norm based one? What is the motivation behind this?

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One motivation of "weak" is, that "strong" convergence implies "weak" convergence. Hence, the "weak" convergence is a weaker convergence than "strong" convergence.

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  • $\begingroup$ To my surprise I discovered that there exists older literature about general topology seeing the concept of "weak" in exactly the opposite way. Have a look at my answer to math.stackexchange.com/q/3234052. This point of view "survived" in the notion of CW-complexes where "W" stands for "weak". $\endgroup$
    – Paul Frost
    Commented Jul 23, 2019 at 13:53
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We speak of weak/strong topologies and also of coarse/fine topologies. Either metaphor relates to the same relationship.

A topology on a set $X$ is a subset $\tau$ of $\mathfrak{P}X$ - the power set of $X$. To qualify as a topology the collection $\tau$ must satisfy the usual axioms. If $\sigma$ is another topology on $X$ then to say $\sigma$ is weaker than $\tau$ means exactly that, as sets $$ \sigma \subseteq \tau. $$ You may find it a useful exercise to explore how this relation induces a partially-ordered set structure on the topologies on $X$, and whether you can find suitable definitions of $\sigma \land \tau$ and $\sigma \lor \tau$ which make this poset into a lattice.

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  • $\begingroup$ so strong vs weak is identical to course vs fine? $\endgroup$ Commented Dec 9, 2015 at 12:42
  • $\begingroup$ yes, with a slight clash of semantics, since coarse corresponds to weak, and strong to fine. however "strong/weak" has taken on a rather more specific family of meanings in the context of topological vector spaces, whereas the "coarse/fine" dimension is of more general application $\endgroup$ Commented Dec 9, 2015 at 13:11
  • $\begingroup$ I see, thank you :) $\endgroup$ Commented Dec 9, 2015 at 13:20

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