The inner product (or ``dot product'', or ``scalar product'') is an operation on two vectors which produces a scalar. Defining an inner product for a Banach space specializes it to a Hilbert space (or ``inner product space''). There are many examples of Hilbert spaces, but we will only need for this book (complex length vectors, and complex scalars).
The inner product between (complex) -vectors and is defined by^{5.9}
The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:^{5.10}
As a result, the inner product is conjugate symmetric:
Note that the inner product takes to . That is, two length complex vectors are mapped to a complex scalar.