(15) 2. Define norm on a function space in terms of vectors (functions) \( \mathbf{U}, \mathbf{V} \) and a scalar \( \alpha \) For the vector space \( R^{3} \), find the uniform norm, the 1-norm and the 2-norm of \( \mathbf{U}=3 \mathbf{i}-4 \mathbf{j}+12 \mathbf{k} \)

The norm on a function space can be thought of as a measure of the "size" or "length" of a function (or vector). For vectors \( \mathbf{U} \) and \( \mathbf{V} \), we can define a norm \( ||\mathbf{U}|| \) that satisfies properties like non-negativity, scalability with \( \alpha \) (i.e., \( ||\alpha \mathbf{U}|| = |\alpha| \cdot ||\mathbf{U}|| \)), and the triangle inequality \( ||\mathbf{U} + \mathbf{V}|| \leq ||\mathbf{U}|| + ||\mathbf{V}|| \). For the vector \( \mathbf{U} = 3 \mathbf{i} - 4 \mathbf{j} + 12 \mathbf{k} \), we calculate the norms as follows: - **Uniform norm (infinity norm)**: \( ||\mathbf{U}||_\infty = \max(|3|, |-4|, |12|) = 12 \). - **1-norm**: \( ||\mathbf{U}||_1 = |3| + |-4| + |12| = 3 + 4 + 12 = 19 \). - **2-norm (Euclidean norm)**: \( ||\mathbf{U}||_2 = \sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \).