Derivative of this application?
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
asked Jan 18 at 1:50
MamanMaman
1,189722
$endgroup$
add a comment |
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
asked Jan 18 at 1:50
MamanMaman
1,189722
$endgroup$
add a comment |
$begingroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
asked Jan 18 at 1:50
MamanMaman
1,189722
$endgroup$
Let consider $xi:mathcal{L}(E)tomathcal{L}(E), f mapsto sum_{nge 0}frac{f^n}{n!}$ where $E$ is a Banach space.
I have to find the expression of the derivative $D_f xi$ for any $hin mathcal{L}(E)$.
Let denote for all $nge 0$ : $xi_n(f)=frac{f^n}{n!}$. Then I try to find $D_fxi_n$ for any $hin mathcal{L}(E)$ by computing $xi_n(f+h)-xi_n(f)$. I start with $n=2,3$ then I want to find an induction formula for all $nge 0$. I found $D_f xi_n(h)=frac{f^{n-1}h+f^{n-2}hf+...+fhf^{n-2}+hf^{n-1}}{n!}$. I do not know if I forgot terms because if there are they are maybe $o(vert{h}vert_{mathcal{L}(E)})$...
Then to get $D_f xi$ should I use a convergence theorem on differentiable sequences in an open bounded convex set of $E$ ?
PS : Apparently we can understand $xi_n$ as a restriction of a $n$-linear map so I was also wondering what kind of informations it brings.
Thanks in advance !
sequences-and-series multivariable-calculus derivatives convergence exponential-function
sequences-and-series multivariable-calculus derivatives convergence exponential-function
asked Jan 18 at 1:50
MamanMaman
1,189722
asked Jan 18 at 1:50
MamanMaman
1,189722
asked Jan 18 at 1:50
MamanMaman
1,189722
asked Jan 18 at 1:50
MamanMaman
1,189722
asked Jan 18 at 1:50
MamanMaman
1,189722
1,189722
add a comment |
add a comment |
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