A sphere net controls a real or complex Hermitian operator norm

Regard $H$ as a complex Hermitian matrix when $\mathbb{K}=\mathbb{R}$. By [F1], it has an orthonormal eigenbasis over $\mathbb{C}$; every eigenvalue is real because $Hv=\lambda v$ and $H=H^{\ast }$ give $\lambda \mathrm{\parallel }v{\mathrm{\parallel }}_2^2=v^{\ast }Hv=\overline{v^{\ast }Hv}=\bar{\lambda }\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Choose a unit eigenvector $z$ for an eigenvalue $\lambda$ of largest absolute value. The diagonalization shows $\mathrm{\parallel }H{\mathrm{\parallel }}_2=\mathrm{\mid }\lambda \mathrm{\mid }$ over $\mathbb{C}$. If $\mathbb{K}=\mathbb{C}$, set $u=z$. If $\mathbb{K}=\mathbb{R}$, write $z=a+ib$; since $H$ and $\lambda$ are real, $Ha=\lambda a$ and $Hb=\lambda b$, and at least one of $a,b$ is nonzero. Normalize a nonzero part as $u$. The same diagonalization bounds the real operator norm by $\mathrm{\mid }\lambda \mathrm{\mid }$, while $u$ attains that bound, so $\mathrm{\mid }{u}^{\ast }Hu\mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\parallel }H{\mathrm{\parallel }}_2$ over either field.

Apply [F2] in real dimension $q=d$ when $\mathbb{K}=\mathbb{R}$, and $q=2d$ when $\mathbb{K}=\mathbb{C}$, to obtain the stated net-size bound and a $v\in \mathcal{N}$ with $\mathrm{\parallel }u-v{\mathrm{\parallel }}_2\le 1\mathrm{/}4$.

Since $H=H^{\ast }$ and $u,v$ are unit vectors, $\mathrm{\mid }{u}^{\ast }Hu-v^{\ast }Hv\mathrm{\mid }\le \mathrm{\mid }(u-v{)}^{\ast }Hu\mathrm{\mid }+\mathrm{\mid }{v}^{\ast }H(u-v)\mathrm{\mid }\le 2\mathrm{\parallel }u-v{\mathrm{\parallel }}_2\mathrm{\parallel }H{\mathrm{\parallel }}_2\le \mathrm{\parallel }H{\mathrm{\parallel }}_2\mathrm{/}2$, using [F3] and the definition of operator norm. Therefore $\mathrm{\mid }{v}^{\ast }Hv\mathrm{\mid }\ge \mathrm{\mid }{u}^{\ast }Hu\mathrm{\mid }-\mathrm{\parallel }H{\mathrm{\parallel }}_2\mathrm{/}2=\mathrm{\parallel }H{\mathrm{\parallel }}_2\mathrm{/}2$, proving the claimed bound.