# Thread: Differentiable mappings and Banach spaces - need help

1. ## Differentiable mappings and Banach spaces - need help

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

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

P.S. How do you define differentiability in Banach spaces?

2. Originally Posted by DeMath
Please, tell me how I can prove this, seemingly, simple statement?

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

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