Linear Functionals Review

Linear Functionals Review

We will now review some of the recent material regarding linear functionals.

  • On the Linear Functionals on Linear Spaces page we said that if $X$ is a linear space then a Linear Functional on $X$ is a linear operator from $X$ to $\mathbb{C}$, and a Real Linear Functional on $X$ is a linear operator from $X$ to $\mathbb{R}$.
  • We then proved an important result. We proved that if $X$ is a linear space and $\varphi \in X^{\#}$ and $x_0 \in X$ is such that $\varphi(x_0) \neq 0$ then:
(1)
\begin{align} \quad X = \ker \varphi \oplus \mathrm{span} (x_0) \end{align}
(2)
\begin{align} \quad \bigcap_{k=1}^{n} \ker \psi_k \subseteq \ker \varphi \end{align}
  • We also noted that $X^{*}$ is always a Banach space.
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