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14
BV-typespaces
1
Herethepreciserequirementson
p
and
λn
willbespecifiedlater.In
thischapterwewilldiscuss,foreachofthesevariations,thealgebraic
andanalyticalpropertiesofthecorrespondingspacesoffunctionsof
boundedvariation,withaparticularemphasisonthoseproperties
whichwillbeimportantinsubsequentchapters.Theproofsofall
statementsgiveninthischapter,togetherwithmanymoreexamples
andremarks,maybefoundinthebook[2].
Thenotationweuseinthisbookisstandard.By
I
wedenotethe
endofaproof.Thesymbol
"x"X
denotesthenormofanelement
x
inanormedspace
X
,whilethesymbol
"A"X→Y
denotestheoperator
normofaboundedlinearoperator
A
betweentwonormedspaces
X
andY.
101FunctionsofboundedJordanvariation
Beforegivingthedefinitionoftheclassicalspaceoffunctionsof
boundedJordanvariation,werecallthedefinitionsofseveralspaces
ofcontinuousfunctionsforfurtheruse.Sincewealwaysconsiderfunc-
tionsovertheinterval
[
0,1
]
,wedropthisintervalinanynotation,i.e.,
wesimplywriteXinsteadofX[0,1]forafunctionspaceX.
By
C
wedenotethelinearspaceofallcontinuousfunctions
x:[0,1]→R,equippedwiththeusualnorm
"x"C:=max{|x(t)||
|0<t<1}(x∈C),
(1.2)
andby
C1
itslinearsubspaceofallcontinuouslydifferentiablefunc-
tionsx:[0,1]→R,equippedwitheitherthenorm
"x"C1:=|x(0)|+"x
!"C(x∈C1)
orthe(equivalent)norm
|||x|||C1:="x"C+"x
!"C.
Moreover,wewillsometimesneedsomeintermediatespacesbetween
C1
and
C
.Recallthatafunction
x:[
0,1
]→R
iscalledLipschitzcontin-
uousifthereexistsaconstantL>0suchthat
|x(s)lx(t)|<L|slt|(0<s,t<1).
(1.3)
