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First: I understand that the completeness relation holds for basis vectors such that ## \sum_{j=1}^{m} | n_{j} \rangle \langle n_{j} | =\mathbb{I}##. Does it also hold for unit-normalized sets of state vectors as well, where ## | \phi_{j} \rangle = c_{j} |n_{j}\rangle ##, because ##\sum_{j=1}^{m} |c_{j}|^2=1 ##, such that ## \sum_{j=1}^{m} | \phi_{j} \rangle \langle \phi_{j} | =\mathbb{I}##? I assume it does not hold for non-unit-normalized sets of state vectors.

Second: For the continuous case, can one define a function ##f(n)=\int_{\mathbb{R}} \phi_{n} dn ##, unit-normalized such that ##\int_{\mathbb{R}} |f(n)|^2 dn =1##, such that ## | f(n) \rangle \langle f(n) | = 1 ##?