How do I prove that <$L(v), v$>$ = 0$ for all $v$ if and only if $L + L$*$ = 0$. Here, $L$ * is the adjoint.
I did something like <$L(v), v$> = <$v, L*(v)$> then added the two to get the $L + L$*, but I don't really know how to proceed.
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Which direction are you having trouble proving? – jgon Nov 09 '18 at 02:07
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1Also what is the context here? Is this a real inner product space? – jgon Nov 09 '18 at 02:09
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Yup, real innter product space. Does <(L+L)(v), v> = 0 for all v automatically imply that L+L is 0? – sedrick Nov 09 '18 at 02:10
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1Have a look at this. – awllower Nov 09 '18 at 02:15
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See the answer here: https://math.stackexchange.com/questions/452802/forall-v-in-vtv-v-0-implies-t-star-t – Dave Nov 09 '18 at 03:32