# Linear dependence of functionals (Intuition)

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 $$\varphi= \sum_{i=1}^{k}\lambda_i\varphi_i$$.

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, $$\{y\}\subset\operatorname{span}\{y_1,\ldots,y_n\}.$$