Mixing
“FTCâ€-type relationship between Frechet Derivative and Bochner integral?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $X,Y$ be Banach spaces and $(X,Sigma,mu)$ be a measure space. Then, for a function $f: Xto Y$ under suitable conditions, we can define the Frechet derivative $text df: Xto B(X,Y)$ or $text df(x): Xto Y$ for some $xin X$, as well as the Bochner integral $int_X f in Y$. They seem to be the natural ways to handle differentiation and integration (resp.) for maps between Banach spaces. This leads me to wonder, is there some kind of connection between the two, akin to FTC?
My attempt was to take a continuous bijective path $gamma$ in $X$ and use some kind of argument similar to FTC for line integrals to get something like $f(x) = f(a) + int_gamma text df(x^*)$, where the measure might be something like $mu([p,q]) = | gamma(q)-gamma(p) |_X $, but I'm not sure this is the right way to go about it.
Otherwise, is there a more natural integration "dual" to the Frechet derivative for maps between Banach spaces for which there is a kind of FTC relationship?
lebesgue-integral banach-spaces frechet-derivative
asked Jul 18 at 15:43
AlexanderJ93
4,067421
add a comment |Â
up vote
0
down vote
favorite
Let $X,Y$ be Banach spaces and $(X,Sigma,mu)$ be a measure space. Then, for a function $f: Xto Y$ under suitable conditions, we can define the Frechet derivative $text df: Xto B(X,Y)$ or $text df(x): Xto Y$ for some $xin X$, as well as the Bochner integral $int_X f in Y$. They seem to be the natural ways to handle differentiation and integration (resp.) for maps between Banach spaces. This leads me to wonder, is there some kind of connection between the two, akin to FTC?
My attempt was to take a continuous bijective path $gamma$ in $X$ and use some kind of argument similar to FTC for line integrals to get something like $f(x) = f(a) + int_gamma text df(x^*)$, where the measure might be something like $mu([p,q]) = | gamma(q)-gamma(p) |_X $, but I'm not sure this is the right way to go about it.
Otherwise, is there a more natural integration "dual" to the Frechet derivative for maps between Banach spaces for which there is a kind of FTC relationship?
lebesgue-integral banach-spaces frechet-derivative
asked Jul 18 at 15:43
AlexanderJ93
4,067421
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X,Y$ be Banach spaces and $(X,Sigma,mu)$ be a measure space. Then, for a function $f: Xto Y$ under suitable conditions, we can define the Frechet derivative $text df: Xto B(X,Y)$ or $text df(x): Xto Y$ for some $xin X$, as well as the Bochner integral $int_X f in Y$. They seem to be the natural ways to handle differentiation and integration (resp.) for maps between Banach spaces. This leads me to wonder, is there some kind of connection between the two, akin to FTC?
My attempt was to take a continuous bijective path $gamma$ in $X$ and use some kind of argument similar to FTC for line integrals to get something like $f(x) = f(a) + int_gamma text df(x^*)$, where the measure might be something like $mu([p,q]) = | gamma(q)-gamma(p) |_X $, but I'm not sure this is the right way to go about it.
Otherwise, is there a more natural integration "dual" to the Frechet derivative for maps between Banach spaces for which there is a kind of FTC relationship?
lebesgue-integral banach-spaces frechet-derivative
asked Jul 18 at 15:43
AlexanderJ93
4,067421
Let $X,Y$ be Banach spaces and $(X,Sigma,mu)$ be a measure space. Then, for a function $f: Xto Y$ under suitable conditions, we can define the Frechet derivative $text df: Xto B(X,Y)$ or $text df(x): Xto Y$ for some $xin X$, as well as the Bochner integral $int_X f in Y$. They seem to be the natural ways to handle differentiation and integration (resp.) for maps between Banach spaces. This leads me to wonder, is there some kind of connection between the two, akin to FTC?
My attempt was to take a continuous bijective path $gamma$ in $X$ and use some kind of argument similar to FTC for line integrals to get something like $f(x) = f(a) + int_gamma text df(x^*)$, where the measure might be something like $mu([p,q]) = | gamma(q)-gamma(p) |_X $, but I'm not sure this is the right way to go about it.
Otherwise, is there a more natural integration "dual" to the Frechet derivative for maps between Banach spaces for which there is a kind of FTC relationship?
lebesgue-integral banach-spaces frechet-derivative
asked Jul 18 at 15:43
AlexanderJ93
4,067421
asked Jul 18 at 15:43
AlexanderJ93
4,067421
asked Jul 18 at 15:43
AlexanderJ93
4,067421
asked Jul 18 at 15:43
AlexanderJ93
4,067421
4,067421
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2855705%2fftc-type-relationship-between-frechet-derivative-and-bochner-integral%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password