1
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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}

$\endgroup$
  • $\begingroup$ very thanks sir. I couldn’t see it in no way :) $\endgroup$ – esrabasar Mar 18 '18 at 10:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.