I'm reading section 6.2 of Gilbarg and Trudinger's book on elliptic PDEs, where the authors write "$\|u\|_{C^{k,\alpha}(\partial\Omega)} = \inf_{\Phi}\|\Phi\|_{C^{k,\alpha}(\overline{\Omega})}$" where $\Phi$ ranges over all extensions of $u$ into $\overline{\Omega}$. However, I don't see how one can prove the triangle inequality with this definition of the norm. Any insight or help is greatly appreciated.
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Suppose $u$ and $v$ are two functions in $C^{1,\alpha}(\partial\Omega)$. For any $\epsilon>0$ there exist extensions $U$ and $V$ such that $$\|U\|_{C^{1,\alpha}(\overline{\Omega})}\le \|u\|_{C^{1,\alpha}(\partial{\Omega})}+\epsilon, \quad \text{and }\ \|V\|_{C^{1,\alpha}(\overline{\Omega})}\le \|v\|_{C^{1,\alpha}(\partial{\Omega})}+\epsilon$$ Then $U+V$ is an extension of $u+v$ and thus $$ \|u+v\|_{C^{1,\alpha}(\partial{\Omega})} \le \|U+V\|_{C^{1,\alpha}(\overline{\Omega})}\le \|u\|_{C^{1,\alpha}(\partial{\Omega})}+ \|v\|_{C^{1,\alpha}(\partial{\Omega})}+2\epsilon$$ Since $\epsilon$ was arbitrarily small, it follows that $$ \|u+v\|_{C^{1,\alpha}(\partial{\Omega})} \le \|u\|_{C^{1,\alpha}(\partial{\Omega})}+ \|v\|_{C^{1,\alpha}(\partial{\Omega})} $$ as required.