10. Inner Product
An inner product is a function that generalizes the dot product to arbitrary vector spaces.
10.1. Axioms
For all and scalar :
- Symmetry:
- Linearity in the first argument:
(by symmetry, also linear in the second argument)
- Positive-definiteness: , with equality only when
A vector space with an inner product is an inner product space.
10.2. Standard examples
Dot product on :
Weighted inner product: where is symmetric positive-definite
Function space inner product (continuous functions on ):
Frobenius inner product on matrices:
10.3. Induced norm
Every inner product induces a norm:
This norm satisfies the parallelogram law — a necessary and sufficient condition for a norm to come from an inner product.
10.4. Cauchy–Schwarz (general form)
The Cauchy–Schwarz inequality holds in every inner product space:
10.5. Orthogonality
Two vectors are orthogonal if their inner product is zero:
This generalizes perpendicularity in . See Orthogonality.
10.6. Angle between vectors
In abstract spaces this defines the angle (between non-zero vectors).
10.7. Why inner products matter
- Generalize geometry (length, angle, projection) to any vector space
- Function spaces: e.g., Hilbert spaces in functional analysis use
- Quantum mechanics: state space is a complex inner product space (Hilbert space)
- Statistics: covariance is an inner product on centered random variables
- Machine learning: kernel methods replace explicit inner products with implicit ones via kernels
10.8. See also
- Dot Product — special case in
- Norm — derived from the inner product
- Orthogonality
- Gram–Schmidt Process — orthogonalization via inner products
- Cauchy–Schwarz Inequality