Let $H$ be a real Hilbert space and $L$ be a Banach limit on $l_{\infty}.$ Denote $S:H \rightarrow H$ as an invertible bounded linear map and there exists positive constants $c$ and $C$ such that $$c\|x\| \leq \|S^nx \| \leq C \| x \|$$ for all $n \in \mathbb{N}$ and $x \in H.$ Define an map $\langle \cdot , \cdot\rangle : H \times H \rightarrow \mathbb{R}$ by $$\langle x,y\rangle = L(((S^nx,S^ny))_{n \in \mathbb{N}})$$ Show that $\langle \cdot,\cdot\rangle $ is an inner product on $H.$
My attempt:
Note that for all $n \in \mathbb{N}$ and $x \in H,$ we have $(S^nx,S^nx) = \| S^n x \|^2 \geq 0.$ Therefore, by the property of Banach limit, it follows that $$\langle x,x\rangle = L(((S^nx,S^nx))_{n \in \mathbb{N}}) = L((\| S^nx\|^2)_{n \in \mathbb{N}}) \geq 0.$$
If $x = 0,$ then $(S^n(0), S^n(0)) = \|S^n(0)\|^2 = \| 0\| = 0$ for all $n \in \mathbb{N}.$ Hence, by the property of Banach limit, $$\langle x,x\rangle = (0,0) = L(((S^n(0),S^n(0)))_{n \in \mathbb{N}}) = 0.$$
If $\langle x,x\rangle = 0 = L(((S^nx,S^nx))_{n \in \mathbb{N}}) = L((\| S^nx \|^2)_{n \in \mathbb{N}}),$ then $\| S^nx\| = 0$ for all $n \in \mathbb{N}.$ By assumption, we have $\| x \| = 0 $, which implies that $x = 0.$
For any $x,y \in H,$ we have $$(x,y) = L(((S^nx,S^ny))_{n \in \mathbb{N}}) = L(((S^ny,S^nx))_{n \in \mathbb{N}}) = (y,x).$$
I do not know how to show $\langle \alpha x + \beta x^{\prime} , y\rangle = \alpha \langle x,y\rangle + \beta \langle x^{\prime}, y\rangle $ and $\langle x, \alpha y + \beta y^{\prime}\rangle = \alpha \langle x,y\rangle + \beta \langle x,y^{\prime}\rangle .$
UPDATE (Bilinearity):
For any $\alpha,\beta \in \mathbb{R}$ and $x,x^{\prime},y\in H$, we have $$(S^n(\alpha x + \beta x^{\prime}), S^n(y)) = (\alpha S^n(x) + \beta S^n(x^{\prime}), S^n(y)) = \alpha (S^n(x), S^n(y)) + \beta (S^n(x^{\prime}), S^n(y)).$$ It follows that \begin{align*} \langle \alpha x + \beta x^{\prime},y \rangle & = L(((S^n(\alpha x + \beta x^{\prime}),S^n(y)))_{n \in \mathbb{N}}) \\ & = L((\alpha (S^n(x), S^n(y)) + \beta (S^n(x^{\prime}), S^n(y)))_{n \in \mathbb{N}}) \\ & = \alpha L(((S^n(x),S^n(y)))_{n \in \mathbb{N}}) + \beta L(((S^n(x^{\prime}),S^n(y)))_{n \in \mathbb{N}}) \\ & = \alpha \langle x , y \rangle + \beta \langle x^{\prime},y \rangle. \end{align*} Similarly, we have $\langle x , \alpha y + \beta y^{\prime} \rangle= \alpha \langle x,y \rangle + \beta \langle x, y^{\prime} \rangle$ for $\alpha, \beta \in \mathbb{R}$ and $x,y,y^{\prime} \in H.$ Therefore, $\langle \cdot, \cdot \rangle$ is an inner product on $H.$
Is my proof correct?