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3
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Definition [ edit ]
Given two separable Banach spaces
E
{\displaystyle E}
and
G
{\displaystyle G}
, a CSM
{
μ
T
|
T
∈
A
(
E
)
}
{\displaystyle \{\mu _{T}|T\in {\mathcal {A}}(E )\}}
on
E
{\displaystyle E}
and a continuous linear map
θ
∈
L
i
n
(
E
;
G
)
{\displaystyle \theta \in \mathrm {Lin} (E;G)}
, we say that
θ
{\displaystyle \theta }
is radonifying if the push forward CSM (see below)
{
(
θ
∗
(
μ
⋅
)
)
S
|
S
∈
A
(
G
)
}
{\displaystyle \left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G )\right\}}
on
G
{\displaystyle G}
"is" a measure, i.e. there is a measure
ν
{\displaystyle \nu }
on
G
{\displaystyle G}
such that
(
θ
∗
(
μ
⋅
)
)
S
=
S
∗
(
ν
)
{\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=S_{*}(\nu )}
for each
S
∈
A
(
G
)
{\displaystyle S\in {\mathcal {A}}(G )}
, where
S
∗
(
ν
)
{\displaystyle S_{*}(\nu )}
is the usual push forward of the measure
ν
{\displaystyle \nu }
by the linear map
S
:
G
→
F
S
{\displaystyle S:G\to F_{S}}
.
Push forward of a CSM [ edit ]
Because the definition of a CSM on
G
{\displaystyle G}
requires that the maps in
A
(
G
)
{\displaystyle {\mathcal {A}}(G )}
be surjective , the definition of the push forward for a CSM requires careful attention. The CSM
{
(
θ
∗
(
μ
⋅
)
)
S
|
S
∈
A
(
G
)
}
{\displaystyle \left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G )\right\}}
is defined by
(
θ
∗
(
μ
⋅
)
)
S
=
μ
S
∘
θ
{\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=\mu _{S\circ \theta }}
if the composition
S
∘
θ
:
E
→
F
S
{\displaystyle S\circ \theta :E\to F_{S}}
is surjective. If
S
∘
θ
{\displaystyle S\circ \theta }
is not surjective, let
F
~
{\displaystyle {\tilde {F}}}
be the image of
S
∘
θ
{\displaystyle S\circ \theta }
, let
i
:
F
~
→
F
S
{\displaystyle i:{\tilde {F}}\to F_{S}}
be the inclusion map , and define
(
θ
∗
(
μ
⋅
)
)
S
=
i
∗
(
μ
Σ
)
{\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=i_{*}\left(\mu _{\Sigma }\right)}
,
where
Σ
:
E
→
F
~
{\displaystyle \Sigma :E\to {\tilde {F}}}
(so
Σ
∈
A
(
E
)
{\displaystyle \Sigma \in {\mathcal {A}}(E )}
) is such that
i
∘
Σ
=
S
∘
θ
{\displaystyle i\circ \Sigma =S\circ \theta }
.
See also [ edit ]
References [ edit ]
t
e
Spaces
Theorems
Operators
Algebras
Open problems
Applications
Advanced topics
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Radonifying_function&oldid=1136988598 "
C a t e g o r i e s :
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H i d d e n c a t e g o r i e s :
● A r t i c l e s l a c k i n g s o u r c e s f r o m D e c e m b e r 2 0 0 9
● A l l a r t i c l e s l a c k i n g s o u r c e s
● P a g e s d i s p l a y i n g w i k i d a t a d e s c r i p t i o n s a s a f a l l b a c k v i a M o d u l e : A n n o t a t e d l i n k
● T h i s p a g e w a s l a s t e d i t e d o n 2 F e b r u a r y 2 0 2 3 , a t 0 4 : 5 0 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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