The Dual of a Real Normed Space Is a Real Banach Space; Weak-Star Limits

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $X$ be a real normed space with dual space $X^{*}$, and for $F\in X^{*}$ let $\lVert F\rVert_{*}$ be its dual norm. For $F,G\in X^{*}$ and $\lambda\in\mathbb{R}$, $F+G$ and $\lambda F$ are the pointwise sum and scalar multiple, and $0$ is the constant zero function on $X$.

(i) Vector space. $F+G\in X^{*}$ and $\lambda F\in X^{*}$ for all $F,G\in X^{*}$ and $\lambda\in\mathbb{R}$, and $X^{*}$ with these operations is a real vector space. Its zero vector is $0$, the negative of $F$ is the function $x\mapsto-F(x)$, and

$$(F-G)(x)=F(x)-G(x)\qquad(F,G\in X^{*},\ x\in X).$$

(ii) Normed space. $\lVert\cdot\rVert_{*}$ is a norm on $X^{*}$.

(iii) Completeness. $(X^{*},\lVert\cdot\rVert_{*})$ is a real Banach space.

Let $(F_{m})_{m\in\mathbb{N}}$ be a sequence in $X^{*}$, and let $F,G\in X^{*}$.

(iv) Uniqueness of weak-star limits. If $(F_{m})$ converges weak-star to $F$ and to $G$, then $F=G$.

(v) Norm convergence implies weak-star convergence. If the real sequence $(\lVert F_{m}-F\rVert_{*})_{m\in\mathbb{N}}$ is a null sequence, that is

$$\lVert F_{m}-F\rVert_{*}\to0,$$

then $(F_{m})$ converges weak-star to $F$.