The inner product is an intuitive geometric concept that is easily introduced for vectors, and it can be used to define a vector norm. A vector norm is a function mapping a vector x into a nonnegative number 
that can be interpreted as vector length. We have see that we may use the dot product to define the usual Euclidean norm:
It turns out that both concepts can be made a bit more abstract and general, and this has a role in generalizing some intuitive concepts such as orthogonality and Pythagorean theorem to quite different fields.12
Generally speaking, the inner product on a space is a function mapping two elements x and y of that space into a real number 
. There is quite some freedom in defining inner products, provided our operation meets the following conditions:
is a real number such that 
and 
only when x = 0.
.
for any scalar α.
.
It is easy to check that our definition of the inner product for vectors in 
meets these properties. If we define alternative inner products, provided they meet the requirements described above, we end up with different concepts of orthogonality and useful generalizations of Pythagorean theorem. A straightforward generalization of the dot product is obtained by considering a vector w of positive weights and defining:
This makes sense when we have a problem with multiple dimensions, but some of them are more important than others.
By a similar token, we may generalize the concept of norm, by requiring the few properties that make sense when defining a vector length:
, with 
if and only if x = 0; this states that length cannot be negative and it is zero only for a null vector.
for any scalar α; this states that if we multiply all of the components of a vector by a number, we change vector length accordingly, whether the scalar is negative or positive.
this property is known as triangle inequality and can be interpreted intuitively by looking at Fig. 3.7 and interpreting vectors as displacement in the plane, and their length as distances. We can move from point P0 to point P1 by the displacement corresponding to vector x, and from point P1 to P2 by displacement y; if we move directly from P0 to P2, the length of the resulting displacement, 
, cannot be larger than the sum of the two distances and 
.
Fig. 3.7 Orthogonal projection of vector u on vector v.
Using the standard dot product, we find the standard Euclidean norm, which can be denoted by 
. However, we might just add the absolute value of each coordinate to define a length or a distance:
The notation is due to the fact that the two norms above are a particular case of the general norm
Letting p → +∞, we find the following norm:
All of these norms make sense for different applications, and it is also possible to introduce weights to assign different degrees of importance to multiple problem dimensions.13
Norms need not be defined on the basis of an inner product, but the nice thing of the inner product is that if we use an inner product to define a norm, we can be sure that we come up with a legitimate norm.
Leave a Reply