# Right topology for infinite dimensional “Hilbert” spaces with indefinite or semidefinite norm

For positive definite infinite dimensional Hilbert spaces, there is the standard Cauchy norm topology. What if this state space has an indefinite norm or a positive semidefinite one, as in gauge theories or Faddeev-Popov ghosts? Which infinite sums are valid, and which aren't?

Similarly, for the algebra of operators, which norm topology do we choose? Not the W*-one? The C* one?

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