[proofplan]
Fix a finite time horizon $\tau > 0$. The essential input is the local uniform boundedness of a strongly continuous semigroup: continuity of each orbit $t \mapsto T(t)x$ on the compact interval $[0,\tau]$, together with the [Uniform Boundedness Principle](/theorems/549), gives a finite bound for $\|T(t)\|_{\mathcal{L}(X)}$ uniformly in $t \in [0,\tau]$. Once this constant is chosen, linearity gives $T(t)u_0 - T(t)v_0 = T(t)(u_0 - v_0)$, and the desired estimate follows from the definition of the operator norm.
[/proofplan]
[step:Choose a uniform operator bound on the finite time interval]
Fix $\tau > 0$. For each $t \in [0,\tau]$, the semigroup operator $T(t)$ belongs to $\mathcal{L}(X)$ by the definition of a strongly continuous semigroup. Define the family of bounded linear operators
\begin{align*}
\mathcal{T}_\tau := \{T(t) \in \mathcal{L}(X) : t \in [0,\tau]\}.
\end{align*}
For each fixed $x \in X$, the orbit map
\begin{align*}
\gamma_x: [0,\tau] &\to X, \quad s \mapsto T(s)x
\end{align*}
is continuous because the semigroup is strongly continuous. Since $[0,\tau]$ is compact and $\gamma_x$ is continuous, the image $\gamma_x([0,\tau])$ is bounded in $X$. Thus
\begin{align*}
\sup_{0 \le s \le \tau} \|T(s)x\|_X < \infty
\end{align*}
for every $x \in X$.
By the Uniform Boundedness Principle applied to the pointwise bounded family $\mathcal{T}_\tau \subset \mathcal{L}(X)$, we obtain
\begin{align*}
\sup_{0 \le s \le \tau} \|T(s)\|_{\mathcal{L}(X)} < \infty.
\end{align*}
This is the standard local boundedness argument for strongly continuous semigroups.
Define
\begin{align*}
M_\tau := \max\left\{1, \sup_{0 \le s \le \tau} \|T(s)\|_{\mathcal{L}(X)}\right\}.
\end{align*}
Then $M_\tau < \infty$ and $M_\tau > 0$.
[guided]
We need one constant that controls all operators $T(t)$ for $0 \le t \le \tau$. The strong continuity assumption gives boundedness along each individual orbit, and the Uniform Boundedness Principle upgrades this pointwise boundedness to a uniform operator-norm bound.
Fix $\tau > 0$ and consider the family
\begin{align*}
\mathcal{T}_\tau := \{T(t) \in \mathcal{L}(X) : t \in [0,\tau]\}.
\end{align*}
This is a family of bounded linear operators on the [Banach space](/page/Banach%20Space) $X$.
We verify pointwise boundedness. Let $x \in X$ be arbitrary. Define
\begin{align*}
\gamma_x: [0,\tau] &\to X, \quad s \mapsto T(s)x.
\end{align*}
Because $(T(t))_{t \ge 0}$ is strongly continuous, $\gamma_x$ is continuous. The interval $[0,\tau]$ is compact, so the continuous image $\gamma_x([0,\tau])$ is compact in $X$, hence bounded. Therefore
\begin{align*}
\sup_{0 \le s \le \tau} \|T(s)x\|_X < \infty.
\end{align*}
Since this holds for every $x \in X$, the family $\mathcal{T}_\tau$ is pointwise bounded.
The Uniform Boundedness Principle applies because $X$ is a Banach space and every member of $\mathcal{T}_\tau$ is a [bounded linear operator](/page/Bounded%20Linear%20Operator) from $X$ to $X$. It gives the operator-norm bound
\begin{align*}
\sup_{0 \le s \le \tau} \|T(s)\|_{\mathcal{L}(X)} < \infty.
\end{align*}
This is the standard local boundedness argument for strongly continuous semigroups.
Now define
\begin{align*}
M_\tau := \max\left\{1, \sup_{0 \le s \le \tau} \|T(s)\|_{\mathcal{L}(X)}\right\}.
\end{align*}
The preceding estimate says that $M_\tau$ is finite, and the maximum with $1$ ensures $M_\tau > 0$. Thus $M_\tau$ is an admissible constant for the finite time interval $[0,\tau]$.
[/guided]
[/step]
[step:Use linearity to reduce the estimate to one initial difference]
Let $u_0, v_0 \in X$ and let $t \in [0,\tau]$. Since $T(t): X \to X$ is linear,
\begin{align*}
T(t)u_0 - T(t)v_0 = T(t)(u_0 - v_0).
\end{align*}
By the definition of the operator norm on $\mathcal{L}(X)$,
\begin{align*}
\|T(t)(u_0 - v_0)\|_X \le \|T(t)\|_{\mathcal{L}(X)} \|u_0 - v_0\|_X.
\end{align*}
Because $t \in [0,\tau]$, the definition of $M_\tau$ gives
\begin{align*}
\|T(t)\|_{\mathcal{L}(X)} \le M_\tau.
\end{align*}
Combining these identities and inequalities yields
\begin{align*}
\|T(t)u_0 - T(t)v_0\|_X \le M_\tau \|u_0 - v_0\|_X.
\end{align*}
[/step]
[step:Conclude continuous dependence uniformly on compact time intervals]
The constant $M_\tau > 0$ depends only on the time horizon $\tau$ and the semigroup $(T(t))_{t \ge 0}$, not on $u_0$, $v_0$, or $t$. Therefore, for every $\tau > 0$, every $u_0, v_0 \in X$, and every $t \in [0,\tau]$,
\begin{align*}
\|T(t)u_0 - T(t)v_0\|_X \le M_\tau \|u_0 - v_0\|_X.
\end{align*}
This is the asserted continuous dependence estimate for semigroup solutions on the finite time interval $[0,\tau]$.
[/step]
