In the book Functional Analysis, Sobolev Spaces and Partial Differential Equations
of Haim Brezis we have the following lemma:
Lemma. Let $X$ be a vector space and let
$\varphi, \varphi_1, \varphi_2, \ldots, \varphi_k$ be $(k + 1)$ linear functionals on $X$ such that
$$
[\varphi_i(v) = 0 \quad \forall\; i \in \{1, 2, \ldots , k\}] \Rightarrow [\varphi(v) = 0].
$$
Then there exist constants $\lambda_1, \lambda_2, \ldots, \lambda_k\in\mathbb{R}$ such that $\varphi=\lambda_1\varphi_1+\lambda_2\varphi_2+\ldots+\lambda_k\varphi_k$.
In this book, the author used separation theorem to prove this lemma. I would like ask whether we can use only knowledge of linear algebra to prove this lemma.
Thank you for all helping.
Best Answer
Your assumption is that $\ker{\varphi} \supseteq \bigcap_{i=1}^k \ker{\varphi_i}$.
Consider the linear map $\ell \colon X \to \mathbb{R}^k$ given by $\ell(x) = (\varphi_1(x),\dots,\varphi_k(x))$ and let $V = \operatorname{im}\ell = \{\ell(x):x \in X\} \subseteq \mathbb{R}^k$ be the image. We have $\ker{\ell} = \bigcap_{i=1}^k \ker{\varphi_{i}} \subseteq \ker\varphi$. Therefore $\varphi = \tilde{\varphi} \circ \ell$ for some linear functional $\tilde{\varphi}\colon V \to \mathbb{R}$ [explicitly, $\tilde{\varphi}(v) = \varphi(x)$ where $x$ is such that $\ell(x) = v$. This is well-defined and linear.]
Every linear functional defined on a subspace $V$ of $\mathbb{R}^k$ can be extended to a linear functional on all of $\mathbb{R}^k$ (write $\mathbb{R}^k = V \oplus V^{\bot}$ and set the extension to be zero on $V^{\bot}$) and every linear functional on $\mathbb{R}^k$ is of the form $\psi(y) = \sum_{i=1}^k a_i y_i$. Thus, there are $\lambda_1,\dots,\lambda_k \in \mathbb{R}$ such that $\tilde\varphi(v) = \sum_{i=1}^k \lambda_i v_i$ for all $v \in V$. In other words, $\varphi = \sum_{i=1}^k \lambda_i \varphi_i$.