Weak-Star Compactness and Vague Convergence: Standing Notation

(i) Base. This setting extends The Dual of C_c(X): Standing Notation: its conventions and notation are in force.

(ii) Balls in a dual space. For a real normed space $X$ with dual space $X^{*}$, $\|F\|_{*}$ denotes the dual norm of $F\in X^{*}$, as in the base, and for a real number $M\ge0$

$$B^{*}_{M}=\{F\in X^{*}:\|F\|_{*}\le M\}.$$

$X^{*}$ carries the pointwise sum and scalar multiplication of The Dual of a Real Normed Space Is a Real Banach Space; Weak-Star Limits (i). Background: The Dual of a Real Normed Space Is a Real Banach Space; Weak-Star Limits.

(iii) Use. The background facts carried by (ii), and their elementary consequences, are used without further citation.