Let X be a real or complex normed space, A be a linear operator in the space X, and x X. We put E(X, A, x) = min{l : l > 0, A l x ≠ x }, or 0 if A κ x = x for all integer κ > 0. Then letE (X, A) = sup x E(X, A, x) andE (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 (overR ), and E(X) ≤ (C p n + p ) 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 - [n] + 2 ≤ E(X) ≤n(n + 1) , 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) ≤ 2ns - 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.