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Let $E$ a vector space and $h,f_1,\cdots,f_n : E \to \mathbb R$ linear such that $\displaystyle \bigcap_{i=1}^n \ker(f_i) \subset\ker h.$ Let $F : E\to \mathbb R^n : x\mapsto(h(x),f_1(x),\cdots,f_n(x))$.

  • Show that $(1,0,\cdots,0)\not \in Im(F)$.
  • Show the exitence of $(\lambda,\lambda_1,\cdots,\lambda_n)\in \mathbb R^{n+1}, \alpha\in \mathbb R$ : $\lambda<\alpha<\lambda h(x)+\sum_{i=1}^n\lambda _if_i(x) \quad\forall x\in E$

The first question is easy to prove bythe absurd but I got stuck in the second one. I know the factorisation of a map by other maps as mentioned here but this exercice is a proof of it . Any help is appreciated.

M-S
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1 Answers1

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Note that $F(E) \subset \mathbb{R}^{n+1}$ is a closed subspace and $e_1 = (1,0,...,0) \notin F(E)$.

Hence the Hahn Banach theorem gives the existence of some linear $\phi$ and some $\alpha$ such that $\phi(e_1) < \alpha < \phi(F(x))$ for all $x$.

Since $\phi$ is linear, it has the form $\phi((h,f_1,...,f_n)) = \lambda h + \sum_k \lambda_k f_k$ from which we get the desired result.

copper.hat
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