Absolutely continuous Banach space valued function The 2019 Stack Overflow Developer Survey...
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Absolutely continuous Banach space valued function
The 2019 Stack Overflow Developer Survey Results Are InWhere does the theory of Banach space-valued holomorphic functions differ from the classical treatment?continuous extension of Banach space-valued analytic functionContinuous function with derivative a.e. and Luzin N property is absolutely continuousAbsolute continuity of a function defined by an integral as $f(x)=int_{0}^{g(x)}h'(t)dt$Validity of simple calculus result for Banach space-valued functions.Can we identify absolutely continuous functions on $(a,b)$ with values in $X$ with $W^{1,2}((a,b);X)$?differentiability of variation of absolutely continuous functionAbsolutely continuous and differentiable almost everywhereLet $f$ be a continuous monotone function. Show that $f$ must be absolutely continuous on [0,1]Is the lebesgue integral of a measurable function continuous?
$begingroup$
Let $X$ be a Banach space and $F:[a,b] to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f in L^1([a,b],X)$, is the function
$$
F(x) = int_{[a,x]}f(t) dt
$$
differentiable almost everywhere and $F'(x) = f(x)$ for almost all $x in [a,b]$? (where the integral is the Lebesgue integral for Banach spaces)
I would also appreciate any literature where this is covered!
functional-analysis banach-spaces absolute-continuity
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space and $F:[a,b] to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f in L^1([a,b],X)$, is the function
$$
F(x) = int_{[a,x]}f(t) dt
$$
differentiable almost everywhere and $F'(x) = f(x)$ for almost all $x in [a,b]$? (where the integral is the Lebesgue integral for Banach spaces)
I would also appreciate any literature where this is covered!
functional-analysis banach-spaces absolute-continuity
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
$endgroup$
$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03
add a comment |
$begingroup$
Let $X$ be a Banach space and $F:[a,b] to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f in L^1([a,b],X)$, is the function
$$
F(x) = int_{[a,x]}f(t) dt
$$
differentiable almost everywhere and $F'(x) = f(x)$ for almost all $x in [a,b]$? (where the integral is the Lebesgue integral for Banach spaces)
I would also appreciate any literature where this is covered!
functional-analysis banach-spaces absolute-continuity
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
$endgroup$
Let $X$ be a Banach space and $F:[a,b] to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f in L^1([a,b],X)$, is the function
$$
F(x) = int_{[a,x]}f(t) dt
$$
differentiable almost everywhere and $F'(x) = f(x)$ for almost all $x in [a,b]$? (where the integral is the Lebesgue integral for Banach spaces)
I would also appreciate any literature where this is covered!
functional-analysis banach-spaces absolute-continuity
functional-analysis banach-spaces absolute-continuity
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
asked Mar 21 at 3:42
Andrei KhAndrei Kh
1,177818
1,177818
$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03
add a comment |
$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03
$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03
add a comment |
1 Answer
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$begingroup$
This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.
answered Mar 21 at 7:30
gerwgerw
20k11334
$endgroup$
add a comment |
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$begingroup$
This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.
answered Mar 21 at 7:30
gerwgerw
20k11334
$endgroup$
add a comment |
$begingroup$
This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.
answered Mar 21 at 7:30
gerwgerw
20k11334
$endgroup$
add a comment |
$begingroup$
This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.
answered Mar 21 at 7:30
gerwgerw
20k11334
$endgroup$
This is not true in arbitrary Banach spaces, but for spaces $X$ possessing the "Radon-Nikodym property". A good resource might be the book "Vector measures" by Diestel and Uhl.
answered Mar 21 at 7:30
gerwgerw
20k11334
answered Mar 21 at 7:30
gerwgerw
20k11334
answered Mar 21 at 7:30
gerwgerw
20k11334
answered Mar 21 at 7:30
gerwgerw
20k11334
20k11334
add a comment |
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$begingroup$
Not certain what you mean by "the Lebesgue integral for Banach spaces". Are you referring to the Bochner Integral for vector measure spaces? You may want to check out Vector Measures by Diestel and Uhl.
$endgroup$
– Theo Bendit
Mar 21 at 3:56
$begingroup$
@Theo Bendit Yes, I did not know that it had a name. But it is defined the same as the usual Lebesgue integral for simple functions but the scalars are elements in a vector space.
$endgroup$
– Andrei Kh
Mar 21 at 4:03