91Ó°ÊÓ

By definition, the absolute value of any real number is non-negative. Therefore, the maximum absolute value of the components of a vector \(\mathbf{x} \in \mathbb{R}^n\) is also non-negative: \[\max _{1 \leq i \leq n}\left|x_{i}\right| \geq 0\] It follows that \(\|\mathbf{x}\|_{\infty} \geq 0\). Moreover, if \(\mathbf{x} = \mathbf{0}\), then all components \(x_i=0\) and \(\|\mathbf{x}\|_{\infty}=0\). Conversely, if \(\|\mathbf{x}\|_{\infty}=0\), then \(|x_i| \le 0\) for all \(1 \leq i \leq n\), which implies that \(x_i = 0\) for all \(1 \leq i \leq n\), and thus \(\mathbf{x} = \mathbf{0}\).

Let \(\alpha\) be a scalar, and let \(\mathbf{x}\) be an element of \(\mathbb{R}^n\). Note that: \[\|\alpha\mathbf{x}\|_{\infty} = \max _{1 \leq i \leq n}\left|\alpha x_{i}\right|=|\alpha| \max _{1 \leq i \leq n}\left|x_{i}\right| = |\alpha|\|\mathbf{x}\|_{\infty}\] Thus, the given expression satisfies the scalability property.

Let \(\mathbf{x}\) and \(\mathbf{y}\) be elements of \(\mathbb{R}^n\). We have: \[\|\mathbf{x}+\mathbf{y}\|_{\infty} = \max _{1 \leq i \leq n}\left|(x_{i}+y_{i})\right| \leq \max _{1 \leq i \leq n}\left(|x_{i}| + |y_{i}|\right)\] We know that \(\max _{1 \leq i \leq n}\left(|x_{i}| + |y_{i}|\right) \leq \max _{1 \leq i \leq n}\left|x_{i}\right| + \max _{1 \leq i \leq n}\left|y_{i}\right|\) because the maximum value of the sum of components is less than or equal to the sum of the maximum values. Hence: \[\|\mathbf{x}+\mathbf{y}\|_{\infty} \leq \|\mathbf{x}\|_{\infty} + \|\mathbf{y}\|_{\infty}\] This proves that the triangle inequality holds for the given expression. Since all three properties (non-negativity, scalability, and triangle inequality) are satisfied, we conclude that the given expression defines a valid norm on \(\mathbb{R}^{n}\).