1. The problem is to understand the function that takes values from the product space in a normed space. 2. A product space of two normed spaces $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ is the set $X \times Y = \{(x,y) : x \in X, y \in Y\}$. 3. We define a norm on this product space by a function such as $$\|(x,y)\| = \sqrt{\|x\|_X^2 + \|y\|_Y^2}$$ or $$\|(x,y)\| = \|x\|_X + \|y\|_Y,$$ which satisfies the norm properties. 4. Important rules for norms: - Non-negativity: $\|z\| \geq 0$ and $\|z\|=0$ iff $z=0$. - Triangle inequality: $\|z_1 + z_2\| \leq \|z_1\| + \|z_2\|$. - Homogeneity: $\|\alpha z\| = |\alpha| \|z\|$ for scalar $\alpha$. 5. The function that takes values from the product space to the normed space is the norm function defined above. 6. For example, if $x \in X$ and $y \in Y$, then the norm of $(x,y)$ in $X \times Y$ is $$\|(x,y)\| = \sqrt{\|x\|_X^2 + \|y\|_Y^2}.$$ This combines the norms of each component into a single value. 7. This function is useful because it allows us to treat pairs $(x,y)$ as elements of a normed space, enabling analysis and operations on product spaces. Final answer: The function is the norm on the product space defined by $$\|(x,y)\| = \sqrt{\|x\|_X^2 + \|y\|_Y^2}$$ or $$\|(x,y)\| = \|x\|_X + \|y\|_Y.$$