The Inner Product

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Inner Product

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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 a Hilbert space (or “inner product space”). There are many examples of Hilbert spaces, but we will only need $\ for this course (complex length $N$ vectors with complex scalars).

The inner product between two (complex) $N$-vectors $x$ and $y$is defined by

\

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:

\

As a result, the inner product is conjugate symmetric:

\

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

Example:For $N=3$ we have, in general,

\

Let

\



Then

\



Subsections

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