CG obeys the Chebyshev $A$-norm bound in terms of the spectral condition number ${\kappa }_2(A)$

Every vector in $x_0+K_m(A,r_0)$ has the form $x_0+Az(A)e_0$ for some polynomial $z$ of degree at most $m-1$, because $r_0=A{e}_0$. Hence every attainable error has the form $e_0-Az(A)e_0=p(A)e_0,$ where $p(0)=1$ and $\mathrm{deg}p\le m$. By [L1], the actual CG error therefore satisfies $\mathrm{\parallel }{e}_m{\mathrm{\parallel }}_A\le {\min }_{\begin{array}{c}p(0)=1\\ \mathrm{deg}p\le m\end{array}}\mathrm{\parallel }p(A)e_0{\mathrm{\parallel }}_A\mathrm{.}$

Since $A$ is Hermitian, [L2] and [L3] give an orthonormal basis of eigenvectors, so $A=U\mathrm{\Lambda }{U}^{\ast }$ with $U$ unitary and $\mathrm{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, where each ${\lambda }_j>0$. Writing $U^{\ast }{e}_0=c$, one gets $\mathrm{\parallel }p(A)e_0{\mathrm{\parallel }}_A^2=⟨Ap(A)e_0,p(A)e_0⟩={\sum }_{j=1}^n{\lambda }_j\mathrm{\mid }p({\lambda }_j){\mathrm{\mid }}^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\le {\max }_{\lambda \in [{\lambda }_{\min },{\lambda }_{\max }]}\mathrm{\mid }p(\lambda ){\mathrm{\mid }}^2\mathrm{\parallel }{e}_0{\mathrm{\parallel }}_A^2\mathrm{.}$ Thus $\mathrm{\parallel }{e}_m{\mathrm{\parallel }}_A\le {\min }_{\begin{array}{c}p(0)=1\\ \mathrm{deg}p\le m\end{array}}{\max }_{\lambda \in [{\lambda }_{\min },{\lambda }_{\max }]}\mathrm{\mid }p(\lambda )\mathrm{\mid }\mathrm{\parallel }{e}_0{\mathrm{\parallel }}_A\mathrm{.}$

Let $t(\lambda ):=\frac{{\lambda }_{\max }+{\lambda }_{\min }-2\lambda }{{\lambda }_{\max }-{\lambda }_{\min }},p_m(\lambda ):=\frac{T_m(t(\lambda ))}{T_m\text{\hspace{0.17em}⁣}\left(\frac{{\lambda }_{\max }+{\lambda }_{\min }}{{\lambda }_{\max }-{\lambda }_{\min }}\right)},$ where $T_m$ is the $m$th Chebyshev polynomial. Then $p_m(0)=1$, $\mathrm{deg}{p}_m=m$, and $\mathrm{\mid }{T}_m(t)\mathrm{\mid }\le 1$ on $[-1,1]$, so ${\max }_{\lambda \in [{\lambda }_{\min },{\lambda }_{\max }]}\mathrm{\mid }{p}_m(\lambda )\mathrm{\mid }\le \frac{1}{\mid T_m\text{\hspace{0.17em}⁣}\left(\frac{{\lambda }_{\max }+{\lambda }_{\min }}{{\lambda }_{\max }-{\lambda }_{\min }}\right)\mid }\mathrm{.}$ With $\kappa ={\lambda }_{\max }\mathrm{/}{\lambda }_{\min }$, the denominator equals $T_m((\kappa +1)\mathrm{/}(\kappa -1))$. For $s>1$, $T_m(s)=\frac{(s+\sqrt{s^2-1}{)}^m+(s-\sqrt{s^2-1}{)}^m}{2}\ge \frac{(s+\sqrt{s^2-1}{)}^m}{2},$ and at $s=(\kappa +1)\mathrm{/}(\kappa -1)$ one has $\frac{1}{s+\sqrt{s^2-1}}=\frac{\sqrt{\kappa }-1}{\sqrt{\kappa }+1}\mathrm{.}$ Therefore ${\max }_{\lambda \in [{\lambda }_{\min },{\lambda }_{\max }]}\mathrm{\mid }{p}_m(\lambda )\mathrm{\mid }\le 2{\left(\frac{\sqrt{\kappa }-1}{\sqrt{\kappa }+1}\right)}^m\mathrm{.}$ Combining this polynomial with step 2.1 gives the stated bound.