Absolutely continuous and differentiable almost everywhere
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I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
$endgroup$
add a comment |
$begingroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
$endgroup$
add a comment |
$begingroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
$endgroup$
I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it:
"A strongly absolutely continuous function which is differentiable
almost everywhere is the indefinite integral of strongly integrable
derivative"
It was in the context of Bochner integrable functions so I'm assuming that "strongly" means with respect to the norm.
Thanks!
integration derivatives bochner-spaces absolute-continuity
integration derivatives bochner-spaces absolute-continuity
asked Sep 20 '18 at 18:21
user202542user202542
144311
144311
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1 Answer
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$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
$endgroup$
add a comment |
$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
$endgroup$
add a comment |
$begingroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
$endgroup$
An absolute continuous function $f:[a,b] to X$ is differentiable almost everywhere such that for a fixed $y in [a,b]$
$$f(x) = int_x^y f'(z) , text{d} z + f(y) quad text{for all } x in [a,b].$$
In particular $f'$ is strongly integrable, i.e. $f' in L^1(a,b;X)$.
You can find this in the book "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. Also in the book "Linear Functional Analysis" by Alt.
answered Dec 6 '18 at 13:02
MarvinMarvin
2,5363920
2,5363920
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