What does a real-linear endomorphism mean?

Question

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If $V$ is a complex $n$-dimensional vector bundle, let $J:V\to V$ denote the complex structure of $V$- the real-linear endormorphism obtained by multiplying with $\sqrt{-1}$

So I am assuming each $v\in V$ is being multiplied with $i$. How is this a "real-linear endomorphism" then? I don't really know what that phrase means, but I would assume it means multiplying with a real number

Answer 1

I would interpret “real linear endomorphism” to mean “the $\mathbb R$ linear transformation of $V\to V$ given by left multiplication by $i$.”

Left multiplication by an element of $\mathbb C$ is already $\mathbb C$ linear, so it is $\mathbb R$ linear as well.

Answer 2

The phrase real-linear endomorphism refers to an $\mathbb{R}$-module homomorphism. This simply means a map satisfying: $\mathit{f}(\mathbf{x+y)}=\mathit{f(x)+f(y)}$ and $\mathit{f(rx)=rf(x)}$