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I'm trying to understand why the linear dependence theorem of functionals is true at an intuitive level. I know the proof given in Brezis's functional analysis book (lemma 3.2), where the Hahn-Banach theorem is used; despite all formal details of the proof are clear to me, I feel I don't understand the intuition underground.

Could you help me?

Linear Dependence Theorem: Let $X$ be a vector space and let $\varphi, \varphi_1,..., \varphi_k$ be $(k+1)$ linear functionals on $X$ such that $ \bigcap_{i=1}^{k}ker(\varphi_i) \subseteq ker(\varphi)$. Then there exist constants $\lambda,\lambda_1,...,\lambda_k$ in $\mathbb{R}$ such that $\varphi= \sum_{i=1}^{k}\lambda_i\varphi_i$.

John Mars
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1 Answers1

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It's kind of natural to see when $X$ is a Hilbert space. Each $\varphi_j$ is of the form $$\varphi_j(x)=\langle x,y_j\rangle,\qquad \varphi(x)=\langle x,y\rangle. $$ So $$\ker\varphi_j=\{y_j\}^\perp,\qquad \ker\varphi=\{y\}^\perp.$$ Then the condition becomes $$\{y\}^\perp\supset\bigcap_j\{y_j\}^\perp=(\operatorname{span}\{y_1,\ldots,y_n\})^\perp.$$ Taking orthogonals, $$\{y\}\subset\operatorname{span}\{y_1,\ldots,y_n\}.$$

Martin Argerami
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