Let $X$ be a normed space. (i) If $x_n \rightharpoonup x$ in $X$, then $\sup_n \|x_n\| < \infty$ and $\|x\| \leq \liminf_{n \to \infty} \|x_n\|$. (ii) If $f_n \overset{*}{\rightharpoonup} f$ in $X^*$ and $X$ is a Banach space, then $\sup_n \|f_n\| < \infty$ and $\|f\| \leq \liminf_{n \to \infty} \|f_n\|$.

Let $(X, \mathcal{P})$ be a locally convex space and $f: X \to \mathbb{F}$ a linear map. Then $f \in X^*$ if and only if $\ker(f)$ is closed.

Every separable normed space $X$ is isometrically isomorphic to a subspace of $\ell_\infty$.

Let $A$ be a unital Banach algebra and let $x, y \in A$ be commuting elements (i.e.\ $xy = yx$). Then \begin{align*} r_A(x + y) &\leq r_A(x) + r_A(y),\\ r_A(xy) &\leq r_A(x)\, r_A(y). \end{align*}

Let $X$ be a normed space. If $X^*$ is separable, then $X$ is separable.

Let $T \in \mathcal{L}(X, Y)$. Then $T^* \in \mathcal{L}(Y^*, X^*)$ and $\|T^*\| = \|T\|$. That is, the map $* : \mathcal{L}(X, Y) \to \mathcal{L}(Y^*, X^*)$ is isometric. Moreover: (i) $(\operatorname{id}_X)^* = \operatorname{id}_{X^*}$; (ii) $(\lambda S + \mu T)^* = \lambda S^* + \mu T^*$ for all $S, T \in \mathcal{L}(X, Y)$ and scalars $\lambda, \mu$; (iii) $(S \circ T)^* = T^* \circ S^*$ for $S \in \mathcal{L}(Y, Z)$, $T \in \mathcal{L}(X, Y)$ (the dual reverses composition).

Let $K$ be compact Hausdorff. For every $\varphi \in M(K) = C(K)^*$, there exists a unique regular complex Borel measure $\nu$ on $K$ such that \begin{align*} \varphi(f) = \int_K f \, d\nu \quad \text{for all } f \in C(K). \end{align*} Moreover $\|\nu\|_1 = \|\varphi\|$, and the map $\varphi \mapsto \nu$ is an isometric isomorphism \begin{align*} C(K)^* \cong \bigl(\{\text{regular complex Borel measures on } K\}, \ \|\cdot\|_1\bigr), \end{align*} where the codomain is equipped with the total-variation norm $\|\nu\|_1 := |\nu|(K)$. If $\varphi \in M_\mathbb{R}(K) = C_\mathbb{R}(K)^*$, then $\nu$ is a signed measure.

Let $X$ be a normed space. (i) A linear functional $f : X \to \mathbb{R}$ is $w$-continuous if and only if $f \in X^*$. Thus $(X, w)^* = X^*$. (ii) A linear functional $\varphi : X^* \to \mathbb{R}$ is $w^*$-continuous if and only if $\varphi = \hat{x}$ for some $x \in X$. Thus $(X^*, w^*)^* = \hat{X}$. (iii) $X$ is reflexive if and only if $\sigma(X^*, X) = \sigma(X^*, X^{**})$, i.e. the weak and weak-$*$ topologies on $X^*$ coincide.

Let $X$ be a normed space. (i) $A \subset X$ is weakly bounded if and only if $A$ is norm-bounded. (ii) If $X$ is a Banach space, then $B \subset X^*$ is $w^*$-bounded if and only if $B$ is norm-bounded.

Let $X$ be a normed space and $K$ a compact Hausdorff space. Then: (i) $X$ is separable if and only if $(B_{X^*}, w^*)$ is metrizable. (ii) $C(K)$ is separable if and only if $K$ is metrizable.