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i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an operator $H$ (Hamiltonian) in the countably-normed Hilbert space $\Phi$, equipped with norms $\left\lVert \cdot \right\rVert_n$. It finishes with an inequality. Point is i dont understand what is the general definition of continuity for an operator in a countably-normed Hilbert space. Does anybody know its definition and could hint me a good textbook where to read the related chapter? Thanks in advance, I hope i phrased this well :)

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