What is meant by
the integral is interpreted in the weak sense
in following corollary:
on page 261 of the book "An introduction to frame and Riesz bases", second edition, by Ole Christenson.
There is no an explanation in the book.
2026-03-27 03:59:48.1774583988
what is meant by "the integral is interpreted in the weak sense"?
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An integral is a limit. A limit depends on the topology you use. In a Hilbert space as $L^2(\mathbb R)$, the two most natural topologies are the strong one, where $f_j\to f$ means $\langle f_j-f,f_j-f\rangle\to0$ (that is, the one given by the norm), and the weak one, where $f_j\to f$ means $$ \langle f_j-f,g\rangle\to0\ \ \ \ \text{ for all } g\in L^2(\mathbb R). $$