91Ó°ÊÓ

To show that \(f = g\) in \(\mathcal{F}\) if and only if \(f = g\) as linear functionals and linear operators, we first define the equality of two linear functionals and two linear operators. Linear functionals are equal if \(f(x) = g(x)\) for all \(x \in V\), while linear operators are equal if \(f(cx) = g(cx)\) for all \(x \in V\) and \(c \in \mathbb{F}\). Assuming \(f = g\) in \(\mathcal{F}\), we show that \(f = g\) as linear functionals by showing \(f(x) = g(x)\) for all \(x \in V\). Similarly, we show that \(f = g\) as linear operators by showing \(f(cx) = g(cx)\) for all \(x \in V\) and \(c \in \mathbb{F}\). Conversely, assuming \(f = g\) as linear functionals and linear operators, we show that \(f = g\) in \(\mathcal{F}\) by using the equality conditions derived for linear functionals and linear operators. Hence, the statement holds in both directions, proving that \(f = g\) in \(\mathcal{F}\) if and only if \(f = g\) as linear functionals, which holds if and only if \(f = g\) as linear operators.