[proofplan] We prove uniqueness by contradiction-free comparison. Given two solutions with the same initial datum, linearity of the homogeneous problem makes their difference a solution with zero initial datum. The a priori estimate then forces the norm of this difference to vanish at every time. Since a norm vanishes only on the zero vector, the two solutions agree pointwise on $[0,T]$. [/proofplan] [step:Subtract two solutions with the same initial datum] Fix $x_0\in X$, and let $u_1,u_2\in\mathcal{S}_{x_0}$ be two solutions. Define the map \begin{align*} w:[0,T]\to X,\qquad t\mapsto u_1(t)-u_2(t). \end{align*} By the assumed linearity of the homogeneous Cauchy problem, $w\in\mathcal{S}_0$. In particular, \begin{align*} w(0)=0. \end{align*} [/step] [step:Apply the energy estimate to the zero initial difference] Since $w\in\mathcal{S}$, the assumed estimate applies to $w$. Thus, for every $t\in[0,T]$, \begin{align*} \|w(t)\|_X\leq C\|w(0)\|_X. \end{align*} Using $w(0)=0$, we obtain \begin{align*} \|w(t)\|_X\leq C\|0\|_X=0. \end{align*} Because norms are nonnegative, this gives $\|w(t)\|_X=0$ for every $t\in[0,T]$. [guided] The point of introducing $w$ is that the estimate controls a solution entirely by its initial value. Since $u_1$ and $u_2$ start from the same datum $x_0$, their difference starts from zero. The map \begin{align*} w:[0,T]\to X,\qquad t\mapsto u_1(t)-u_2(t) \end{align*} belongs to $\mathcal{S}_0$ by the linearity hypothesis. Therefore it is a solution to which the a priori estimate may be applied, and its initial value is \begin{align*} w(0)=u_1(0)-u_2(0)=x_0-x_0=0. \end{align*} Applying the estimate to this specific solution $w$ gives, for every $t\in[0,T]$, \begin{align*} \|w(t)\|_X\leq C\|w(0)\|_X. \end{align*} Substituting $w(0)=0$ yields \begin{align*} \|w(t)\|_X\leq C\|0\|_X=0. \end{align*} A norm is always nonnegative, so the inequality $\|w(t)\|_X\leq 0$ forces \begin{align*} \|w(t)\|_X=0. \end{align*} This is the exact place where the energy estimate becomes a uniqueness tool: zero initial difference cannot grow into a nonzero difference at later times. [/guided] [/step] [step:Conclude pointwise equality of the two solutions] For each $t\in[0,T]$, the norm identity $\|w(t)\|_X=0$ implies $w(t)=0$ by definiteness of the norm on $X$. Hence, for every $t\in[0,T]$, \begin{align*} u_1(t)-u_2(t)=0. \end{align*} Therefore $u_1(t)=u_2(t)$ for every $t\in[0,T]$, so $u_1=u_2$ as maps $[0,T]\to X$. Since $x_0\in X$ was arbitrary, each initial datum admits at most one solution in the class for which the estimate holds. [/step]