Linear Functionals Review
• 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:
• We also noted that $X^{*}$ is always a Banach space.