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Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement...enter image description here

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement... 

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement...

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement...enter image description here

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all coactions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

I have no idea on how to prove this statement...

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement...