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*complete*, with

*completeness*in this sense heuristically meaning that all possible sequences of elements within this space have a well-defined limit that is itself an element of this space (I think this is right?!). This is a useful property as it enables one to do calculus in this space.

Now, in quantum mechanics Hilbert spaces play an important role in that they are the spaces in which the (pure) states of quantum mechanical systems "live". Given a set of orthonormal basis vectors, ##\lbrace\lvert\phi_{n}\rangle\rbrace## for such a Hilbert space, one can express a given state vector, ##\lvert\psi\rangle## as a linear combination of these basis states, $$\lvert\psi\rangle=\sum_{n}c_{n}\lvert\phi_{n}\rangle$$ since the basis states are orthonormal, i.e. ##\langle\phi_{n}\lvert\phi_{m}\rangle =\delta_{nm}## we find that ##c_{n}=\langle\phi_{n}\lvert\psi\rangle##, and hence $$\lvert\psi\rangle=\sum_{n}c_{n}\lvert\phi_{n}\rangle =\sum_{n}\langle\phi_{n}\lvert\psi\rangle\lvert\phi_{n}\rangle =\left(\sum_{n}\lvert\phi_{n}\rangle\langle\phi_{n}\lvert\right)\lvert\psi\rangle$$ which implies that $$\sum_{n}\lvert\phi_{n}\rangle\langle\phi_{n}\lvert =\mathbf{1}$$ This is referred to as a

*completeness relation*, but I'm unsure what this is referring to? I've also read that the basis must be complete. Is this referring to the notion of completeness associated with limits of sequences, or is there something else I'm missing? And also, apart from being implied in the "derivation" I did above, why does completeness of a given basis require that the sum of outer products of each basis vector with itself equals the identity?