Triangle Inequality Trouble in a Norm

Question

I know it should be very easy but I have really stuck

$X$ and $Y$ are vector spaces and $\|\cdot\|_1$ is a norm on $X$ and $\|\cdot\|_2$ is a norm on $Y$.

Show that $\|(x,y)\| = \max\{\|x\|_1,\|y\|_2\}$ is a norm on $X \times Y$.

I don’t have any problem about first three of them but I cannot continue in triangle inequality

What did I do:

Let $x_1,x_2 \in X$ and $y_1,y_2 \in Y$

I think I have to show that,

$\|(x_1+x_2,y_1+y_2)\| \le \max \{\|x_1\|_1,\|y_1\|_2\} + \max \{\|x_2\|_1,\|y_2\|_2\}$

\begin{align} \|(x_1+x_2,y_1+y_2)\| &= \max \{\|x_1+x_2\|_1,\|y_1+y_2\|_2 \}\\ &\le \max \{\|x_1\|_1+ \|x_2\|_1,\|y_1\|_2+ \|y_2\|_2\}\\ &\le \|x_1\|_1+ \|x_2\|_1+\|y_1\|_2+ \|y_2\|_2 \end{align}

What should I do about inequalities? I guess there are some mistakes or unecessary things

Could someone help me please

Thanks

Answer 1

Using the triangle inequality for $\|\cdot\|_1$ and $\|\cdot\|_2$ we get

$$\|x_1 + x_2\|_1 \le \|x_1\|_1 + \|x_2\|_1 \le \max \{\|x_1\|_1,\|y_1\|_2\} + \max \{\|x_2\|_1,\|y_2\|_2\}$$

$$\|y_1 + y_2\|_2 \le \|y_1\|_2 + \|y_2\|_2 \le \max \{\|x_1\|_1,\|y_1\|_2\} + \max \{\|x_2\|_1,\|y_2\|_2\}$$

So we conclude

\begin{align} \|(x_1, y_1) + (x_2, y_2)\| &= \max\{\|x_1 + x_2\|_1, \|y_1 + y_2\|_2\} \\ & \le \max\{\|x_1\|_1,\|y_1\|_2\} + \max \{\|x_2\|_1,\|y_2\|_2\} \\ &= \|(x_1, y_1)\| + \|(x_2, y_2)\| \end{align}