Motivated by the well-known inequality concerning the Fourier coefficients \(a_ k(f)\) and \(b_ k(f)\) of a function \(f\in L^ \infty_{2\pi}\), the author considers pseudomodulars \(\rho_ 1,\rho_ 2,\rho_ 3\) satisfying the inequality (I) \(\rho_ 3(cx)<C\rho_ 1(x)\rho_ 2(x)\) defined on a vector space \(X\), where \(c\) and \(C\) are positive constants, and proves the following.
If the above inequality (I) is satisfied for every \(x\) of \(X\), then the identity operator in \(X\) is a continuous map from \(\langle X,\rho_ 1,\rho_ 2\rangle\) to \(\langle X,\rho_ 3\rangle\).
After giving a proof of the above, the author gives a number of cases in analysis for which the above inequality (I) is satisfied.