Let X be a real or complex normed space, A be a linear operator in the space X, and x ϵ X. We put E( X, A, x) = min{ l : l>0, ∥ A l x∥ ≠ ∥ x∥}, or 0 if ∥ A k x∥ = ∥ x∥ for all integer k>0. Then let E( X, A) = sup x E( X, A, x) and E( X) = sup A E( X, A). If dim X ≥ 2 then E( X) ≥ dim X + 1. A space X is called E-finite if E( X) < ∞. In this case dim X < ∞, and we set dim X = n. The main results are following. If X is polynomially normed of a degree p, then it is E-finite; moreover, E( X) ≤ C p n+ p−1 (over R), and E(X) ≤ (C p 2 n+p 2−1 ) 2 (over C). If X is Euclidean complex, then n 2 − n + 2 ≤ E( X) ≤ n 2 − 1 for n ≥ 3; in particular, E( X) = 8 if n = 3. Also, E( X) = 4 if n = 2. If X is Euclidean real, then [ n 2 ] 2 − [ n 2 ] + 2 ≤ E(X) ≤ n(n + 1) 2 , and E( X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E( X, A) ≤ 2 ns − s 2, where s is the number of nonzero eigenvalues. For any operator A we prove that E( X, A) ≤ n 2 − n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are “small” and can be found exactly. For instance, E( X, A) ≤ 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.