# 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.

• Positive operators need not be injective. $B=0$ is also a positive operator. Jul 15 '21 at 9:39
• It depends if a positive operator is allowed to have a kernel. My definition allows it! And if I want a positive operator not to have a kernel, I would say a "positive definite operator", I guess.
– Plop
Jul 15 '21 at 9:41
• Yes, it's rather a terminology question, 'positive operator' here means positive semidefinite. Jul 15 '21 at 9:43
• Yeah thanks I was pretty sure that it didnt admit a kernel. But checking better it does Jul 15 '21 at 9:55
• yes right @Berci with positive I immidiately meant positive semidefinite, but it's semidefinite what it was meant Jul 15 '21 at 10:03

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.