.
.(The inner product is linear in the first variable, and conjugate linear in the second.)
Sequences with the property that
are called Cauchy sequences. Usually one works
with Hilbert spaces because one needs to have available such limits of Cauchy sequences. Finite dimensional inner product spaces are automatically Hilbert spaces.
However, it is the infinite dimensional Hilbert spaces that are important
for the proper foundation of quantum mechanics.
A Hilbert space is also a Banach space in the norm induced by the inner product, because both the norm and the inner product induce the same metric.
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