Theorem (Hölder’s inequality): For \(1/p + 1/q = 1\) and \(\x, \y \in \real^d\),

\[\x \Tr \y \leq \norm{\x}_p\norm{\y}_q,\]

with exact equality iff \(\x = a\y\) for some scalar \(a\) (unless \(p\) or \(q\) is exactly 1).