Seminorm ($p$)

A seminorm on a vector space is a nonnegative function $p$ satisfying $p(\lambda x)=|\lambda |p(x)$ and $p(x+y)\le p(x)+p(y)$. Unlike a norm, it may vanish at nonzero vectors.

Given a bilinear pairing $⟨\cdot ,\cdot ⟩$ and a seminorm $p$, its dual seminorm isIt may be infinite when $y$ does not annihilate the kernel of $p$. Whenever it is finite, the defining inequality gives $|⟨x,y⟩|\le p(x)p^{\ast }(y)$.