Expressing a Lin. Functional as a Linear Comb. of Other Lin. Functionals
Expressing a Linear Functional as a Linear Combination of Other Linear Functionals
If $X$ is a linear space and $\varphi, \psi_1, \psi_2, ..., \psi_n$ are linear functionals, we would like to obtain a criterion for when $\varphi$ is a linear combination of $\psi_1, \psi_2, ..., \psi_n$.
The theorem below gives us such a criterion.
Theorem 1: Let $X$ be a linear space and let $\varphi, \psi_1, \psi_2, ..., \psi_n \in X^{\#}$. Then $\varphi$ is a linear combination of $\psi_1, \psi_2, ..., \psi_n$ if and only if $\displaystyle{\bigcap_{k=1}^{n} \ker \psi_k \subseteq \ker \varphi}$. |
We only prove the forward direction of the theorem above. The converse is rather complicated so we omit it.
- Proof: $\Rightarrow$ Suppose that $\varphi$ is a linear combination of $\psi_1, \psi_2, ..., \psi_n$. Then there exists $a_1, a_2, ..., a_n \in \mathbb{C}$ such that:
\begin{align} \quad \varphi(x) = a_1\psi_1(x) + a_2\psi_2(x) + ... + a_n\psi_n(x) \quad (*) \end{align}
- Let $x \in \bigcap_{k=1}^{n} \ker \psi_k$. Then $\psi_k(x) = 0$ for all $k \in \{ 1, 2, ..., n \}$. Therefore $\varphi(x) = 0$ from $(*)$. So:
\begin{align} \quad \bigcap_{k=1}^{n} \ker \psi_k \subseteq \ker \varphi \quad \blacksquare \end{align}