[proofplan] The upper bound follows immediately from the Peano kernel representation by pulling out $\|f^{(k+1)}\|_\infty$. Sharpness is proved by constructing a sequence of smooth functions whose $(k+1)$-th derivatives approximate $\operatorname{sgn}(K_L)$, which nearly saturates the bound. [/proofplan] [step:Derive the upper bound from the Peano kernel] By the [Peano Kernel Theorem](/theorems/484), \begin{align*} |L(f)| &= \frac{1}{k!}\left|\int_a^b K_L(\theta)\,f^{(k+1)}(\theta)\,d\mathcal{L}^1(\theta)\right| \leq \frac{\|f^{(k+1)}\|_\infty}{k!}\int_a^b |K_L(\theta)|\,d\mathcal{L}^1(\theta) = c_L\,\|f^{(k+1)}\|_\infty. \end{align*} [/step] [step:Prove sharpness by approximating $\operatorname{sgn}(K_L)$] For any $\varepsilon > 0$, let $g_\varepsilon \in C[a,b]$ approximate $g = \operatorname{sgn}(K_L)$ with $\|g - g_\varepsilon\|_{L^1} < \varepsilon$. Define $f_\varepsilon$ by integrating $g_\varepsilon$ a total of $k+1$ times so that $f_\varepsilon^{(k+1)} = g_\varepsilon$ and $\|f_\varepsilon^{(k+1)}\|_\infty = 1$. Then \begin{align*} |L(f_\varepsilon)| &= \frac{1}{k!}\left|\int_a^b K_L(\theta)\,g_\varepsilon(\theta)\,d\mathcal{L}^1(\theta)\right| \geq \frac{1}{k!}\left(\int_a^b |K_L|\,d\mathcal{L}^1 - \|K_L\|_\infty \cdot \varepsilon\right). \end{align*} Since $\varepsilon$ is arbitrary, $|L(f_\varepsilon)| > c_L - \delta$ for any $\delta > 0$, confirming sharpness. [/step]