3.11. UNIVERSAL APPROXIMATION 127
as theStone-Weierstrass theorem, provides the most general framework for
designing function approximators.
To formulate this theorem, we need some terminology and concepts. LetX
be a set. AdistanceormetricdonXis a mapd:X◊X→Rsuch that
1.d(x,y)≥ 0 andd(x,y)=0if and only ifx=y.2.d(x,y)=d(y,x).- For anyx,y,z∈X,d(x,y)≤d(x,z)+d(z,y).
OnR, absolute valued(x,y)=|x−y|is a metric, and on the setC([a,b])
of all continuous real-valued functions definedontheclosedinterval[a,b],the
function
d(f,g)= sup
x∈[a,b]
|f(x)−g(x)|forf,g∈C([a,b])is also a metric. This metric is generated from thesup-norm
kfk=sup
x∈[a,b]|f(x)|by lettingd(f,g)=kf−gk.
Since fuzzy systems or neural networks produce functions fromRntoR,we
need to specify the most general form of the counterpart inRnof an interval
[a,b]inR.Solet(X,d)beametricspace,thatis,asetX together with a
metricdon it. For example, the Euclidean distance forX=Rnis
d(x,y)=≥P
n
i=1(xi−yi)2¥^12
forx=(x 1 ,...,xn)andy=(y 1 ,...,yn)AsubsetAofXis said to beopenif for eacha∈Athere is anε> 0 such
that{x∈X:d(x,a)<ε}⊆A. A subsetBofXis said to beclosedif its
set complementB^0 ={x∈X:x/∈B}is open. A subsetAofXisbounded
ifsup{d(x,y):x,y∈A}<+∞.OnRn, subsets that are closed and bounded
are calledcompact subsets. In metric spaces(X,d),asubsetAofXis said
to becompactif any open cover of it contains afinite subcover ñ that is, for
any open setsAi,i∈I, such thatA⊆∪i∈IAi,thereisafinite setJ⊆Isuch
thatA⊆∪j∈JAj. The closed intervals[a,b]inRare compact, and this is the
property that we need to generalize toRn.
Let(X,d)and(Y,e)be two metric spaces. A functionf:X→Y is said
to becontinuous at a pointx∈Sif for anyε> 0 ,thereisδ> 0 such that
e(f(x),f(y))<εwheneverd(x,y)<δ.ThespaceC([a, b])of continuous
real-valued functions on[a, b]is generalized to the spaceC(X)of continuous
real-valued functions onX,where(X,d)is a metric space andXis compact.
Thesup-normonC(X)issup{|f(x)|:x∈X}.
By exploiting the properties of polynomials on[a,b], we arrive at the Stone-
Weierstrass theorem. Its proof can be found in any text in Real Analysis, for
example see [61].