Let X be a complex Hilbert space with the inner product $\langle⋅,⋅\rangle$ and norm $||⋅||$. Suppose T is a bounded linear operator on $X$ and $T$ is positive definite, means there is a constant $K>0$, such that $K\langle x,x\rangle \leq \langle T(x),x\rangle$, prove $T$ has a bounded inverse $T^{-1}$ and the norm of $T^{-1}$ is less than or equal to $K$.
I simply put $K$ into the inner product, then $K\langle x,x\rangle \leq \langle T(x),x\rangle$, is that means $Kx \leq T(x)$? hope get prove of this question