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

Question

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

Any suggestions would be nice.

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 $\rangle$. — Harald Hanche-Olsen, Oct 07 '14 at 14:02

score 8 · Accepted Answer · edited Oct 07 '14 at 14:03

8

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 $u-v=0$.

edited Oct 07 '14 at 14:03

answered Oct 07 '14 at 14:00

agha

See my comment on the question, above. – Harald Hanche-Olsen Oct 07 '14 at 14:03

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