I've just encountered the following theorem : If T is a linear operator in a complex vector space V then if(adsbygoogle = window.adsbygoogle || []).push({});

< v , Tv > =0 for all v in V then T=0

But the theorem doesn't hold in real 2-D vector space as the operator could be the operator that rotates any vector by 90 degrees. My question is : the set of complex numbers includes the set of real numbers so if a theorem holds in a complex vector space then how can it not hold in a real vector space ?

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# Real and complex vector spaces

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