L-infinity embeds continuously into BMO modulo constants

For every cube $Q$ the triangle inequality and [F2] give $\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-b_Q\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }{b}_Q\mathrm{\mid }\le 2\mathrm{\parallel }b{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$, so $b$ is locally integrable and $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le 2\mathrm{\parallel }b{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$; in particular $L^{\mathrm{\infty }}({\mathbb{R}}^n)\subseteq \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$.

Adding a constant $c$ to a representative gives $(b+c{)}_Q=b_Q+c$ by linearity of the integral, so the class map is well defined on $L^{\mathrm{\infty }}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$: both quotients identify functions whose difference is almost everywhere constant. If two classes $b_1,b_2\in L^{\mathrm{\infty }}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$ have the same image in $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$, then $b_1-b_2$ is almost everywhere constant by the definition of that quotient as the quotient of $\mathrm{B}\mathrm{M}\mathrm{O}$ by the constants, so $b_1$ and $b_2$ already represent the same class in $L^{\mathrm{\infty }}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$; that is injectivity.

By invariance of the BMO seminorm under constants and step 1.1, for every $c\in \mathbb{C}$ one has $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}=\mathrm{\parallel }b-c{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le 2\mathrm{\parallel }b-c{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$. Taking the infimum over $c$ gives $\mathrm{\parallel }[b]{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}\mathrm{/}\mathbb{C}}\le 2\mathrm{\parallel }[b]{\mathrm{\parallel }}_{L^{\mathrm{\infty }}\mathrm{/}\mathbb{C}}$ for the quotient norms, so the induced class map is continuous; step 1.2 gives injectivity. Both statements are choice-free: only linearity of the integral and the definition of the seminorm are used.