XVI
General Notation
If x , y G R , the n w e writ e x A y = min(x , y) fo r th e minimu m an d
x V y = max(x , y) fo r th e maximum . Similarly , su p an d in f respectivel y
refer t o supremu m an d infimu m operations .
Functions. I f X an d Y ar e two sets, then " / : X Y" stand s fo r " / map s
X int o Y," an d "x /(#) " refer s t o th e ma p fro m x t o f(x). I f / : X Y
and A C Y , the n f~
1
{A) = {x G X : f(x) G ^4}. Thi s i s the inverse image
of A.
The Big-O/Little- o Notation . Suppos e ai, «2,..., &i,&2 £ R » W e say
that a
n
~ 6
n
[a s n oo ] whe n lim n_*oo(an/6n) = 1. Whe n th e b^s ar e als o
non-negative, "limsup n_^00 \a n\/bn oo " i s ofte n writte n a s "a
n
= 0(6
n
),"
and "lim^-^o o |a
n
|/6
n
= 0 " a s "a
n
= o(6 n)." Not e tha t a
n
= 0(b n) if f ther e
exists a constan t C suc h tha t \a n\ Cb
n
fo r al l n 1.
The big-O/little- o notatio n i s also applicable t o functions :
u
f(x) ~ #(x )
as x - ^ a " mean s
a
limx_^a(/(x)/^(x)) = 1"; an d whe n g 0 w e ma y
write
u
f(x) 0(g(x)) a s x » a" fo r "limsup x_a |/(x) | = 0(#(x)), " an d
"/(#) = o(7(#) ) a s x a " i n plac e o f "lim x-+a(f(x)/g(x)) 0. "
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