Bounded, coercive and symmetric sesquilinear forms

Let $H$ be a real or complex Hilbert space over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ with inner product $(\cdot ,\cdot )$ linear in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length, Hilbert space), and let $a:H\times H\to \mathbb{K}$ be sesquilinear in the sense of Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable: linear in the first argument and conjugate-linear in the second. The form $a$ is bounded with bound $M\ge 0$ when $\mathrm{\mid }a(u,v)\mathrm{\mid }\le M\mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }\text{for all }u,v\in H,$ and coercive with constant $\alpha >0$ when $\mathrm{Re}a(u,u)\ge \alpha \mathrm{\parallel }u{\mathrm{\parallel }}^2\text{for all }u\in H\mathrm{.}$ It is symmetric (Hermitian) when $a(u,v)=\overline{a(v,u)}$ for all $u,v$, over $\mathbb{C}$ this is equivalent to $a(u,u)\in \mathbb{R}$ for every $u$. Indeed, writing $q(u)=a(u,u)$, sesquilinearity gives $4a(u,v)=q(u+v)-q(u-v)+iq(u+iv)-iq(u-iv)$, and real diagonal values make this identity conjugate-symmetric. Over $\mathbb{R}$, symmetry means $a(u,v)=a(v,u)$; real diagonal values alone do not imply symmetry. The adjoint form is $a^{\ast }(u,v):=\overline{a(v,u)}$, and $(a^{\ast }{)}^{\ast }=a$. Real bilinear convention. When $H$ is a real Hilbert space the same definitions apply with $a$ bilinear and coercive in the form $a(u,u)\ge \alpha \mathrm{\parallel }u{\mathrm{\parallel }}^2$; the conjugation in the second slot is then the identity. More generally, a real bilinear form may satisfy the boundedness and coercivity conditions without being symmetric; symmetry is an additional property, not part of either condition. When $H=\{0\}$ every form is bounded with bound $0$ and coercive with every $\alpha >0$; this degenerate case is kept but is never load-bearing. All constants below are named and never silently improved. No choice principle is used in this definition.