# Thread: Norm on a Cartesian product

1. ## Norm on a Cartesian product

Let $V_1, V_2$ be normed vector spaces with norms $||\cdot||_1$ and $||\cdot||_2$, respectively. Show that $(||v_1||_1^p + ||v_2||_2^p)^\frac{1}{p}, 1\leq p < \infty$ defines a norm on the space $V_1\times V_2$.

I've got the first three criteria down (positive definite, zero only when the vector is the zero vector, absolute homogeneity), however I'm having trouble seeing how to break this apart to show it satisfies the triangle inequality.

Suggestions on how to get started?

2. Originally Posted by mathematicalbagpiper
Let $V_1, V_2$ be normed vector spaces with norms $||\cdot||_1$ and $||\cdot||_2$, respectively. Show that $(||v_1||_1^p + ||v_2||_2^p)^\frac{1}{p}, 1\leq p < \infty$ defines a norm on the space $V_1\times V_2$.

I've got the first three criteria down (positive definite, zero only when the vector is the zero vector, absolute homogeneity), however I'm having trouble seeing how to break this apart to show it satisfies the triangle inequality.
Given two elements $(u_1,u_2),\ (v_1,v_2)$ in $V_1\times V_2$, you need to show that the triangle inequality $\bigl(\|u_1+v_1\|_1^p + \|u_2+v_2\|_2^p\bigr)^{1/p} \leqslant \bigl(\|u_1\|_1^p + \|u_2\|_2^p\bigr)^{1/p} + \bigl(\|v_1\|_1^p + \|v_2\|_2^p\bigr)^{1/p}$ holds.

Use the triangle inequality in $V_1$ and $V_2$ to see that the left side of that inequality is $\leqslant \bigl((\|u_1\|_1+\|v_1\|_1)^p + (\|u_2\|_2+\|v_2\|_2)^p\bigr)^{1/p}$. Then use Minkowski's inequality in two-dimensional space to conclude that that is less than or equal to the right side of the triangle inequality.

3. That does work out nicely, except for the fact that if I use that I won't get credit for it, since this exercise is in section 1.2 of the book and Minkowski's Inequality doesn't get introduced until Section 1.5, thus I don't have license to use it.

(My professor is very picky about that).

4. Originally Posted by mathematicalbagpiper
That does work out nicely, except for the fact that if I use that I won't get credit for it, since this exercise is in section 1.2 of the book and Minkowski's Inequality doesn't get introduced until Section 1.5, thus I don't have license to use it.

(My professor is very picky about that).
The general form of Minkowski's inequality may not come until Section 1.5, but are you sure that some special cases of it haven't come earlier? Your problem only needs the inequality for a 2-dimensional space. In fact, if $V_1$ and $V_2$ are both 1-dimensional spaces then the problem is actually equivalent to the 2-dim. Minkowski inequality. So somehow or other you will have to make use of it!

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