4
$\begingroup$

If in an inner product space $\langle x,u\rangle=\langle x,v\rangle$ for all $x$, show that $u=v$.

This seems obvious to me, so how do I prove it? Proof by contradiction maybe?

Any suggestions would be nice.

$\endgroup$
1
  • 3
    $\begingroup$ Please don't use $<$ and $>$ for inner products. Those are relation symbols, and they not only look wrong, they also get the spacing associated with relations. Use \langle and \rangle to get $\langle$ and $\rangle$. $\endgroup$ Oct 7 '14 at 14:02
6
$\begingroup$

Let $x=u-v$, then:

$$\langle x,u\rangle-\langle x,v\rangle=\langle x,u-v\rangle=\langle u-v,u-v\rangle=0$$

So $u-v=0$.

$\endgroup$
1
  • $\begingroup$ See my comment on the question, above. $\endgroup$ Oct 7 '14 at 14:03

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.