A nonzero element $e \in \mathcal{K}(\mathcal{H})$ is called a projection, if it is self adjoint and idempotent. In addition, if $e\mathcal{K}(\mathcal{H})e = \mathbb{C}e$ then, it is called a minimal projection. Suppose that $e_0\in \mathcal{K}(\mathcal{H})$ is a minimal projection and $E_{e_0} =\lbrace xe_0 : x\in E\rbrace$.
Why $E_{e_0}$ is a Hilbert space with the inner product $(xe_0, ye_0) = tr(\langle xe_0, ye_0\rangle)$ for all $x, y\in E$ ?