It turns out that
the di
fference only matters for Hilbert spaces of infinite dimension, in which case there can arise
bra vectors whose corresponding ket vector is of infinite length, i.e. has infinite norm, and hence
cannot be normalized to unity. Such ket vectors can therefore never represent a possible physical
state of a system. But these issues will not be of any concern here. The point to be taken away
from all this is that a bra vector is not the same kind of mathematical object as a ket vector. In fact,
it has all the attributes of an operator in the sense that it acts on a ket vector to produce a complex
number, this complex number being given by the appropriate inner product.
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