Let X be a Hilbert space with inner product denoted $(\cdot,\cdot)$ and norm $\|\cdot\|$ induced by it. Suppose $F\,\colon\,X \times \mathbb{R} \to X$ is continuously Frechet differentiable with respect to both variables. Assume $(x_0,y_0)$ are such that $F(x_0,y_0) = 0$. Define $$ T\,\colon\,X \to X\quad \text{ with }\quad Tx = DF(x_0,y_0)(x,0) $$ and assume that it is an isomorphism, which in particular implies that there exist constants $C,D >0$ such that $$ C\|x\| \leq \|Tx\| \leq D \|x\|.\quad (1) $$ Does it imply that there exists $\lambda >0$ such that $$ (Tx,x) \geq \lambda (x,x)\,\forall x \in X?\quad (2) $$ Am I missing something absolutely obvious? Does (1) straightfowardly imply (2)?
Futhermore, this setup permits us to apply Implicit Function Theorem for Banach spaces (as defined here) to conclude that there is some neighbourhood $U\subset X$ around $x_0$ and some interval $I\subset \mathbb{R}$ around $y_0$ and some Frechet differentiable function $g\,\colon\,\mathbb{R}\to X$ such that $F(g(y),y) = 0$ and $F(x,y) = 0$ iff $x = g(y)$ for all $(x,y) \in U \times I$.
Again, is it obvious that then (2) applies to $g(y_1) \in X$ for any $y_1 \in I?$ Is it a consequence the fact that $x\mapsto D(g(y_1),y_1)(x,0)$ is also an isomorphism? Is it?