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If yes, how can you prove it?

We have $\langle T(v),w\rangle =\langle v,T^*(w)\rangle$. If $T(v)=0$, then what?

  • 3
    If $T(v)=0$, what is $\left$?2012-12-03

2 Answers 2

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Using the definition of adjoint, $ \langle 0^*v,w\rangle = \langle v,0w\rangle=\langle v,0\rangle =0. $ As this happens for every $w$, $0^*v=0$ for all $v$. i.e. $0^*=0$.

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If $T(v)=0$ $\forall v$, this implies $\langle v, T^\ast(w)\rangle=0$ $\forall v$.

In particular, you can choose $v=T^\ast(w)$ and find $\langle T^\ast(w), T^\ast(w)\rangle=0$ and thus $T^\ast(w)=0$.