Proposition I.6

$$ Let $(VV,+,\circ )$ over $𝕂$ be an algebra of endomorphisms of $V$.

• Suppose that $\varphi \in VV$ such that there exists $\phi \in VV$ with $\mathit{\varphi \phi }=$ and $\mathit{\phi \varphi }=$. We show that

  $$\forall <!--\; FUNCTION\; APPLICATION\; -->y\in V\exists <!--\; FUNCTION\; APPLICATION\; -->!x\in V:\varphi (x)=y$$

  Existence: let $y\in V$ be arbitrary. Denote $x:=\phi (y)$. We then have

  $$\varphi (x)=\varphi (\phi (y))=(y)=y$$

  as desired. Now we show uniqueness. Suppose that we have $x,x^{\prime }\in V$ with $\varphi (x)=y$ and $\varphi (x^{\prime })=y$. Then $\phi (y)=\phi (\varphi (x))=x$ and $\phi (y)=\phi (\varphi (x^{\prime }))=x^{\prime }$, and so $x=x^{\prime }$.

• If $\varphi$ is an isomorphism, then by definition ${\varphi }^{-1}$ is its inverse with respect to $\text{∘}$.