A vector space with the inner product space
Also referred to as spatial linear vector space, the vector of the vector space enclosed linear combination. If 
is a set of basis vectors of the space V, it is 
still in the vector space V. In the vector space, only defines a vector multiplication and addition operations. Based on this, the inner product computation defined by the inner product operation, a vector length concept can be solved, and so the angle between the vectors, which define the inner product space. Set vector is X, Y, X is defined as the length 
, X, is defined as the angle between Y 
.
Two inner product
In 
space, the following vector 
and 
, in geometry, a length of the vector represents the distance from the origin to its endpoint, according to the Pythagorean theorem, there 
. The inner product is defined 
, equal to the length of the vector X 
, so that the establishment of the relationship between the length of the inner product.
In the complex vector space 
, the following vectors and the inner product is defined 
. How to understand the complex vector inner product? First of all, for a single complex 
,
There 
, the conjugated complex multiplication can be solved length. When two different complex conjugate multiplication 
, the result is still a complex number, it can be decomposed into a real number and imaginary number component classification. Complex vector inner product of the resulting complex is obtained by adding a result, the end result typically comprises a real component and an imaginary number component portion, i.e., the result is generally 
in the form.
Inner product satisfies the following properties:
1) positive: If v is a nonzero vector, <v, v>> 0, the vector nature of the solid complex vector are true;
2) Conjugate Symmetry: 
for complex vector, equation holds for real vectors, it is equal to conjugate operation itself, the inner product calculation symmetry;
3) Uniformity: 
for when the complex vector c is a complex number, c is a real number real vector;
4) Linear: <u + v, w> = <u, w> + <v, w>, <u, v + w> = <u, v> + <u, w>, for a complex vector with real vectors They have been set up.
Three 
space and 
space
A signal may be expressed as a function f (t) is, in the section 
on spatial 
denotes the space of all square integrable function of the composition, i.e.,
The function f (t) may be present in an infinite number of discontinuous points, using Lebesgue view, i.e. without considering the set of measure zero in the interval 
integral and the limited. In the N-dimensional vector space, a space of dimension N, but also for the vector length N. Analog N-dimensional vector space, the space is infinite dimensional (i.e., an infinite number of f (t) satisfies the above conditions), the interval may be infinitely thin, similar to the vector length can be infinite.
Suppose f (x), g (x ) is a 
signal space, the interval [0, 1] is discretized into N aliquots, where N is a vector constituted 
, 
when N approaches infinity greatly,
, 
Then 
, 
.
When gradually increasing N, 
also gradually increases, since the 
space for the infinite dimensional space, if the definition of the product obtained by this method an infinite value (in the vector space, due to limited space dimension, and use of product definition reasonable physical meaning is very clear). Improved methods for the infinite, and average there 
. When N approaches infinity, this formula is approximate and Riemann, the 
inner product may be defined as:
， 
。
A space to meet the same product positive, conjugate symmetry, uniformity and other properties linear.
Since the Lebesgue integral process, does not consider the discontinuous point measure is zero, then the 
defined space in addition to the two functions are equal to zero means that the measure set, as long as f (t) = g (t ) function that other regions i.e. equal.
In signal processing applications, there are many discrete infinite sequence 
, which sequence is the discrete j> | N | time, 
which defines the 
discrete form:
, As there is no longer defined as an average value is a discrete sequence of j> | N |, the .