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## The Inner Product

The

inner product(or “dot product”) is an operation on two vectors which produces a scalar. Adding an inner product to a Banach space produces aHilbert space(or “inner product space”). There are many examples of Hilbert spaces, but we will only need for this course (complex length vectors with complex scalars).The

inner productbetween two (complex) -vectors and is defined by

The complex conjugation of the second vector is done in order that a

normwill beinducedby the inner product:

As a result, the inner product isconjugate symmetric:

Note that the inner product takes to . That is, two length complex vectors are mapped to a complex scalar.

Example:For we have, in general,

Let

Then

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