1.1. Vecto r Space s
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times a continuou s functio n i s continuous , w e hav e tha t C[a, b] is a
vector spac e ove r th e scala r field R .
By th e sam e reasoning , th e se t o f al l complex-value d continuou s
functions o n [a , b] is a vecto r spac e ove r th e comple x scala r field C .
The same notation C [a, b] i s usually use d to denote this complex func -
tion space , wit h th e particula r scala r field specifie d b y th e contex t i n
which th e notatio n i s used i f there i s n o ambiguity .
This notatio n ca n b e generalize d t o othe r domains . Fo r instance ,
C(R) wil l denot e th e se t o f al l continuou s function s o n R .
Example 1.4. A functio n spac e whic h w e don' t ofte n tal k abou t
is th e se t F(D) o f al l functions , continuou s o r not , fro m a specifie d
domain D int o th e scala r field F . Th e functio n spac e F(D) ha s th e
mathematically interestin g propertie s o f bein g maximal, i n th e sens e
that i t i s not containe d a s a proper subse t o f any larger functio n spac e
on th e domai n D, an d als o universal, i n tha t i t contains , a s subsets ,
all othe r functio n space s o n D. Thes e ar e particularl y interestin g
features whe n w e not e that , amon g th e mor e abstrac t spaces , ther e
are n o maxima l vecto r space s an d n o universa l vecto r spaces .
Example 1.5. Anothe r functio n spac e tha t w e wil l frequentl y en -
counter i s th e se t F
n
[a, b] of al l polynomial s o n th e interva l [a , b] o f
degree les s than o r equa l t o n , fo r a specifie d positiv e intege r n , wit h
coefficients i n F . Similarly , P
n
(R) denote s th e se t o f al l polynomial s
on R o f degre e n o r les s wit h coefficient s i n F . A s above , th e scala r
field i s usuall y specifie d b y context . I t i s clea r tha t thes e set s ar e
closed unde r additio n an d scala r multiplication , s o the y ar e indee d
vector spaces .
From th e abov e examples , th e reade r shoul d no t b e surprise d t o
find that , i n general , th e mos t attractiv e an d usefu l functio n space s
are thos e tha t ar e proper subsets o f th e maxima l functio n space s
F(D). Thi s i s ofte n tru e i n othe r vecto r space s a s well . Whil e ther e
do not exis t maxima l vector space s in general, man y vector spaces ar e
often bes t studie d a s prope r subset s o f large r vecto r space s i n whic h
they ar e naturall y contained , o r embedded . Th e reade r shoul d loo k
for thi s phenomeno n a s w e describ e mor e example s o f vecto r spaces .
This lead s u s t o th e definitio n o f a subspac e o f a vecto r space .
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