**The Unit Operator**

Consider a complete orthonormal system (*basis*), for a Hilbert space *H*, with respect to the norm from an inner product . From basic functional analysis we know that any ket can be written as

with the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now follows that

must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting its value, for example

where in the last identity Einstein summation convention has been used.

In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients and of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to insert the unit operator into the bracket one time or more (for more information see Resolution of the identity).

Read more about this topic: Bra-ket Notation

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