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Math Help - Differentiable mappings and Banach spaces - need help

  1. #1
    Senior Member DeMath's Avatar
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    Differentiable mappings and Banach spaces - need help

    Please, tell me how I can prove this, seemingly, simple statement?

    If \langle{E,\|\!\cdot\!\|_1}\rangle,\,\langle{F,\|\!  \cdot\!\|_2}\rangle - the Banach spaces, G\subset{E} - an open set and mappings G\xrightarrow{f_1}F,\,G\xrightarrow{f_2}F are differentiable at the point x_0\in{G}, then the mappings g_1=f_1+f_2 and g_2=\lambda{f},\,\lambda\in\mathbb{R} also differentiable at the point x_0, and with equalities dg_1(x_0)=df_1(x_0)+df_2(x_0),~dg_2(x_0)=\lambda{d  }f_1(x_0).


    P.S. How do you define differentiability in Banach spaces?
    Last edited by DeMath; May 1st 2010 at 11:27 AM.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by DeMath View Post
    Please, tell me how I can prove this, seemingly, simple statement?

    If \langle{E,\|\!\cdot\!\|_1}\rangle,\,\langle{F,\|\!  \cdot\!\|_2}\rangle - the Banach spaces, G\subset{E} - an open set and mappings G\xrightarrow{f_1}F,\,G\xrightarrow{f_2}F are differentiable at the point x_0\in{G}, then the mappings g_1=f_1+f_2 and g_2=\lambda{f},\,\lambda\in\mathbb{R} also differentiable at the point x_0, and with equalities dg_1(x_0)=df_1(x_0)+df_2(x_0),~dg_2(x_0)=\lambda{d  }f_1(x_0).

    P.S. How do you define differentiability in Banach spaces?
    Probably the Fréchet derivative is what is wanted.
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