What is the difference between strong and weak convergence?
I am reading "Introductory functional analysis" by Kreyszig and I dont appreciate the differences between the two.
Definition of strong convergence:
A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x \in X$ such that $$\lim_{n \to \infty}||x_n-x||=0$$
Definition of weak convergence:
A sequence $(x_n)$ in a normed space $X$ is said to be weakly convergent if there is an $x \in X$ such that $$\lim_{n \to \infty}f(x_n)=f(x)$$
I do not appreciate the differences between the two, does anyone have an example to highlight the differences?
How does the proof differ in showing if a sequences converges weakly
or strongly?
Best Answer
It all boils down to realizing that it is natural (and interesting!) to consider different topologies on the same set $X$, each of which comes with a notion of convergence.
In this scenario we have on one hand the strong topology, i.e. the topology induced by norm on $X$, and the so called weak topology on the other, i.e. the coarsest topology for which every element in $X^*$ is continuous.
The first thing to realize is that $\tau_{\text{weak}} \subset \tau_{\text{strong}}$. Indeed,
The other implication does not hold as shown by the following example:
Here you can find a proof that for finite dimensional spaces the two topologies coincide.
It is worth mentioning it is not true that the topologies are different if and only if $X$ is infinite dimensional. For a counterexample you can take a look at this question ($\ell^1$ is a weird space).