Separated seminormed space

Question

Let $(F, \langle \cdot\;,\;\cdot\rangle)$ be a complex Hilbert space. Let $M$ a positive semidefine operator on $F$.

Clearly $(F,\|\cdot\|_M)$ is a seminormed space, where $$\|x\|_M=\langle Mx\;,\;x\rangle^{1/2},$$ for all $x\in F$.

Clearly $(F,\|\cdot\|_M)$ is a normed space iff $M$ is injective.

Assume that $M$ is not injective, is $(F,\|\cdot\|_M)$ a separated space? My goal is to see if the limit of a given convergent sequence in $(F,\|\cdot\|_M)$ is unique or not.

Answer 1

If $Mx =0$ with $x \neq 0$ then $\{x,x,x...\}$ converges to both $x$ and $0$.

[If $x_n=x$ for all $n$ then $\|x_n-x\|=\|0\|=0$ so $x_n \to x$. Also, $\|x_n -0\|=\langle Mx , x \rangle =0$ since $Mx=0$. Hence $x_n \to 0$ also].