Separable Real Normed Spaces and Hahn–Banach: Notation

(i) Base. This setting extends Real Normed, Inner Product and Hilbert Spaces: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force.

(ii) Balls and subsets. For a real normed space $X$ with norm metric $d_X$, a point $x\in X$ and a real number $r>0$, $B(x,r)$ denotes the open ball $B_{d_X}(x,r)$. Open subsets of $X$ refer to $(X,d_X)$, and closed and compact subsets of $X$ refer to the topological space $(X,\mathcal{T}_{d_X})$.

(iii) Countable sets. Countable sets are as in Countable Set.

(iv) Background. An inequality between real numbers that holds up to every positive slack holds exactly, by Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (i) and Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (ii); and the supremum of a nonempty subset of $\mathbb{R}$ that is bounded above, or the infimum of one that is bounded below, is approached within every positive $\varepsilon$ by elements of the subset, by Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (v) and Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers (vii). These facts are used without further citation.