Is it a norm or a seminorm? Let $X$ Hilbert space and $B \in L(X)$ positive, self adjoint operator. Then is this a norm or seminorm?
$$|x|_B=(\langle Bx,x\rangle)^{1/2}$$
In  [Li, Xungjing, and Jiongmin Yong. Optimal control theory for infinite dimensional systems.1995] p.232 they claim it's a seminorm.
But since $B$ is positive I can define $B^{1/2}$ which is a positive self adjoint operator and then
$$|x|_B=(\langle Bx,x\rangle )^{1/2}=(\langle B^{1/2}x,B^{1/2}x\rangle )^{1/2}=|B^{1/2}x|$$
which is zero only for $x=0$ by the positivity of $B^{1/2}$.
What am I missing?
The thing that come to mind is that of course $\langle Bx,x\rangle =0$ when $Bx$ is orthogonal to $x$ but this seems not to be the case thanks to the previous equality.

As already explained by @Plop and @Berci, it depends on whether $B$ is strictly positive definite or only positive semidefinite.
For a positive semidefinite $B$, you may have some $v \in X$, $ v \neq 0 $ with $Bv = 0$. Then also $ |x|_B = 0 $ and $ |\cdot |_B $ this is only a seminorm, not a norm.
For a strictly positive definite $ B $, also $ B^{1/2} $ is strictly positive definite (by spectral calculus), so $ x \neq 0 $ implies $ |B^{1/2}x| > 0 $ and
$$ |x|_B = |B^{1/2}x| > 0, $$
so you indeed have a norm.
