Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 179

sensors) in terms of a finite number of basis functions ϕ 1 ϕ m , which are gen-

erally nonlinear functions of the plant measured variables (secondary variables).
The model is simply obtained by projecting y into the space spanned by the linear
combination of these bases. This is so whether y is a vector in a Euclidean space,
a function in an inner product function space, a random variable in an inner prod-
uct random variable space, a random sequence in a random sequence inner product
space, etc.
For the purposes pursued in this chapter an inner product space ℵis a linear
space in which a real valued function called inner product is definedonanytwoof
its elements x

, ϕ 2 , ...,

, y . The inner product is denoted as x , y ,so

x , y : ℵ → ℜ . (4.76)

The axioms that an inner product must satisfy are the same as the ones for an Eu-
clidean space. This is why other inner product spaces “look like” a Euclidean space.

4.3.1.1 Inner Product Axioms

The four axioms an inner product must satisfy are [63, 64]

( IP 1 ) x + z , y = x , y + z , y ,

( IP 2 ) α x , y = α x , y , α ∈ ℜ ,

(4.77)

( IP 3 ) x , y = y , x ,

( IP 4 ) x , x > 0 , x = 0 , x , x = 0 ⇐⇒ x = 0 .

It may easily be seen that in the Euclidean space

x T y x , y = , (4.78)

also known as dot product, is an inner product, because it satisfies these four axioms.
Two elem en ts x , y in an inner product space are said to be orthogonal if x , y = 0

y is also used in general. Orthogonality defined as x T y and the notation x ⊥ = 0ina

Euclidean space is a particular case.
Just as in Euclidean spaces, the “size” of an element x is its norm , and

2

x = x , x . (4.79)

This is also a general property of inner products and this norm is called the norm
induced by the inner product. The norm assigns to each element a real number ac-
cording to (4.79). The four axioms for the norm in a normed linear space - and also
in an inner product space, where the norm is the induced norm - are [63, 64]

( N 1 ) x ≥ 0 ,

( N 2 ) x + y ≤ x + y ,

(4.80)

( N 3 ) α x = α x ,

( N 4 ) x = 0 ⇐⇒ x = 0 .