= Strong continuity of functional calculus through resolvents
{title2=$a_i\to a\text{ strongly}\Longrightarrow f(a_i)\to f(a)\text{ strongly},\ f\in C_0(\mathbb R)$}
For bounded <self-adjoint> $a_i,a$, no uniform bound on $\|a_i\|$ is needed. The <resolvent identity> gives $(a_i-i)^{-1}-(a-i)^{-1}=(a_i-i)^{-1}(a-a_i)(a-i)^{-1}$, and the first factor has <norm> at most one. Thus both <resolvents> at $i$ and $-i$ converge strongly. Products preserve strong convergence for uniformly bounded nets; uniform approximation by the algebra of these <resolvent> functions then gives the displayed statement for every continuous function vanishing at infinity.
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