An inner product is a linear operator often used to test constituents of a vector subspace for orthogonality. The inner product, while applying to geometric, real, and complex vectors, and functions, still abides by four general rules. They are as listed by ariels:
Linearity 1: <x|y+z> = <x|y> + <x|z>
Linearity 2: <x|ky> = < kx|y> = k<x|y> for all k real
Symmetry: <x|y> = <y|x>
Non-negativity: <x|x> = ||x||||x|| >= 0 (where ||x|| denotes the norm of x which is zero iff x is a zero function or the zero vector)
The inner product as defined for geometric, real vectors is also known as the dot product or scalar product. For two geometric vector, real vectors, x and y, the inner product can be written: <x|y> = ||x||||y||cos(a) or = x1y1 + x2y2 + . . . + xnyn
In other words, the inner product or the vectors x and y is the product of the magnitudes of the vectors times the cosine of the non-reflexive (<=180 degrees) angle between them. Or the inner product of x and y is the sum of the products of each component of the vectors.
For real or complex n-tuples, the definition is changed slightly. For two real or complex n-tuples x and y, < x|y > = < x*|y > = ||x||*||y||cosa = x1*y1 + x2*y2 + . . . + xn*yn (here * denotes the complex conjugate not multiplication which is implied anyways.)
In other words, the inner product of two real or complex n-tuples is the product of the magnitude of the second n-tuple with the magnitude of the complex conjugate of the first n-tuple. Or the inner product of two real or complex n-tuples x and y is the sum of the products of the complex conjugates of the components of the first n-tuple and the product of the components of the second n-tuple. The definition is pretty much the same for n-tuples and geometric vectors except that in the complex case one n-tuple gets evaluated as its complex conjugate. Because of the symmetry principle, it doesn't matter which n-tuple, but by convention the first one is chosen. This need to use the complex conjugate for only one n-tuple arises from the need to preserve the non-negativity of the norm. If x is complex, ||x||||x|| = ||x||^2 may be negative.
Finally, for functions, the inner product is defined much differently. Note also that the definition is only valid for piecewise continuous functions.
For two piecewise continuous functions f and g on a closed interval a to b, = ∫fgdx (the limits of integration are from a to b).
The norm a function is also defined as ||f|| = >= 0. Hence we can define the inner product redundantly as = ||f||||g||cosa. It is difficult to find a geometric analogy for this definition but it should be stressed that the norm of a function does not represent its arc length and the angle a does not represent the angle between the functions.
The best use for the inner product is to define and test for orthogonality. For geometric vectors, orthogonality and perpendicularity may be considered the same thing but this definition does not hold for complex n-tuples or functions. A better definition is that two geometric vectors, real, or complex n-tuples, or functions are orthogonal if and only if their inner product is zero. This is extremely important as it makes the inner product the only condition for orthogonality, a very important and useful concept throughout linear algebra.