Fortin operators transfer the continuous inf-sup bound

The case $Q_h=\{0\}$ is vacuous, so fix $q_h\mathrm{\ne }0$. In these real spaces, replacing a nonzero velocity by its negative shows that the supremum of $b(v,q_h)\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_V$ equals the supremum with $\mathrm{\mid }b(v,q_h)\mathrm{\mid }$ in the numerator, both on $V$ and on any nonzero subspace $V_h$. The continuous inf-sup bound gives ${\sup }_{v\mathrm{\ne }0}\mathrm{\mid }b(v,q_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_V\ge \beta \mathrm{\parallel }{q}_h{\mathrm{\parallel }}_Q>0$, so some $v$ has $b(v,q_h)\mathrm{\ne }0$. Constraint preservation [F1] then implies ${\mathrm{\Pi }}_{h}v\mathrm{\ne }0$, and its norm bound forces $C_F>0$; in particular $V_h\mathrm{\ne }\{0\}$. Define $s(q_h):={\sup }_{v_h\mathrm{\ne }0}\mathrm{\mid }b(v_h,q_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V$. For every nonzero $v\in V$ with ${\mathrm{\Pi }}_{h}v\mathrm{\ne }0$, [F1] and the definition of this fixed-$q_h$ supremum give $\mathrm{\mid }b(v,q_h)\mathrm{\mid }=\mathrm{\mid }b({\mathrm{\Pi }}_{h}v,q_h)\mathrm{\mid }\le s(q_h)\mathrm{\parallel }{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_V\le C_{F}s(q_h)\mathrm{\parallel }v{\mathrm{\parallel }}_V$. If ${\mathrm{\Pi }}_{h}v=0$, preservation gives $b(v,q_h)=0$, so the same bound holds.

Taking the supremum over nonzero $v$ in step 1.1 and using the continuous bound yields $\beta \mathrm{\parallel }{q}_h{\mathrm{\parallel }}_Q\le C_{F}s(q_h)$. Divide by the positive $C_F\mathrm{\parallel }{q}_h{\mathrm{\parallel }}_Q$, and only then take the infimum over $q_h\mathrm{\ne }0$. The real-sign equivalence in step 1.1 identifies ${\inf }_{q_h\mathrm{\ne }0}s(q_h)\mathrm{/}\mathrm{\parallel }{q}_h{\mathrm{\parallel }}_Q$ with the stated ${\beta }_h$, giving ${\beta }_h\ge \beta \mathrm{/}{C}_F$.