Dual Operator is Well-Defined

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be normed vector spaces over $\GF$.

Let $T : X \to Y$ be a bounded linear transformation.

Let $X^\ast$ and $Y^\ast$ be the normed duals of $X$ and $Y$ respectively.

Then the dual operator $T : Y^\ast \to X^\ast$ is well-defined.

Proof
We first show that $T^\ast$ is well-defined as a mapping $Y^\ast \to X^\ast$.

That is, we want to show that $T^\ast f \in X^\ast$ for each $f \in Y^\ast$.

For $x, y \in X$ and $\lambda, \mu \in \GF$, we have:

So $T^\ast f$ is a linear functional.

We show that $T^\ast f$ is bounded.

For each $x \in X$, we have:

So $T^\ast f$ is bounded.