6
1. Linea r Algebr a Revie w
(i) x + y = y + x.
(ii) (x + y) + z = x + (y + z) .
(hi) x + z = y ha s a uniqu e solutio n z fo r eac h pai r (x , y).
(iv) a(/3x) = (a(3)x.
(v) ( a - f /?) x = ax + /?# .
(vi) a( x + y) = ax + ay.
(vii) l x = x.
Elements o f a vecto r spac e ar e calle d vectors. Fo r an y vecto r
space V , ther e exist s a uniqu e vecto r z suc h tha t z + x x fo r al l
x G V. Thi s vector z i s called the zero vector of V and wil l be denote d
by 0 . Th e trivial vecto r spac e i s the spac e {0 } consisting o f th e zer o
vector alone . Anothe r typica l exampl e o f a vecto r spac e i s Euclidea n
space
F n : =
with th e usua l additio n an d scala r multiplication .
Many natura l example s o f vecto r space s ar e collection s o f func -
tions unde r th e usua l operation s o f multiplicatio n b y scalar s an d ad -
dition o f functions. Tha t is , if / an d g are functions o n some specifie d
domain D, an d i f a G F , then af an d f + g are defined t o be the func -
tions o n D give n b y (af)(x) = af(x) an d ( / + g)(x) = f(x) + g(x),
respectively, fo r al l point s x G D. I t i s straightforwar d t o sho w tha t
any se t o f functions tha t i s closed unde r thi s definitio n o f scalar mul -
tiplication an d additio n satisfie s al l th e vecto r spac e axioms , s o suc h
a se t i s indee d a vecto r space .
Definition 1.2. A function space is a set o f functions o n some speci-
fied domain tha t i s closed unde r functio n additio n an d multiplicatio n
by scalars .
Example 1.3. Le t [a , 6] be a close d bounde d interva l i n R . Defin e
C[a, b] to b e th e se t o f al l continuou s real-value d function s o n [a , b].
Since the su m o f two continuous function s i s continuous, an d a scala r
Xi
J^n
\
:
Xi
G F
)
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