The Inner Product

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.

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Subsections

• • • Example:
  
  
• Linearity of the Inner Product
• Norm Induced by the Inner Product
• Cauchy-Schwarz Inequality
• Triangle Inequality
• Triangle Difference Inequality
• Vector Cosine
• Orthogonality
• The Pythagorean Theorem in N-Space
• Projection

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.