The strong convergence of a sequence of bounded linear operators $T_n:X\to Y$ , where $X,Y$ are Banach spaces, is the pointwise convergence $T_n x \to T x$ for all $x \in X$. The weak star convergence of a sequence $f_n$ in the topological dual $X'$ of $X$ is defined as $f_n(x) \to f(x)$ for all $x \in X$. Is it just the strong convergence of bounded linear operators $f_n:X \to \mathbb{R}$? If so, why do we need a separate name for the convergence? Is it just a matter of terminology or is there some significant difference that we cannot use the term strong convergence?
A confusion on the strong convergence of operators and weak star convergence of functionals
banach-spacesfunctional-analysisoperator-theory
Best Answer
The two notions are equivalent, as you have observed. $X'$ has its weak* topology and weak* convergence of a sequence $\{f_n\}$ in $X'$ is just convergence in this topology. Many books on FA discuss weak and weak* topology but don't discuss enough of operator theory to introduce strong convergence of operators. Thus weak* convergence is more basic and that is the reason this concepts should not be forgotten.