TITLE: Does a mathematical proof in $\mathbb{C}$ imply a proof in the $\mathbb{R}$?
QUESTION [11 upvotes]: Does every proof in the complex numbers also prove the statement in the real numbers? I thought it might be true, because the real numbers are part of the complex numbers.

REPLY [6 votes]: There is a problem in Kreyszig's Functional Analysis:
Let $X$ be an inner product space over $\mathbb{C}$, and $T:X\to X$ is a linear map. If $\langle x,Tx\rangle =0\:\forall x\in X$, then $T$ is the null transformation.
This is not true for inner product spaces over  $\mathbb R$.