Under the assumptions that Δ(f, h)(t) = |f(t + h) − f(t)|, X is a symmetric space of functions in [0, 1], α ∈ (0, 1) and p ∈ [1, ∞) are any fixed number, by the triple (X, α, p) a Besov type space Λ X,p α is constructed, where the norm is given by the equality $$ \left\| {f\left| {\Lambda _{X,p}^\alpha } \right.} \right\| = \left( {\sum\limits_{i = 1}^\infty {(2^{\alpha i} \left\| {\Delta (f;2^{ - 1} )( \cdot )|X} \right\|)^p } } \right)^{1/p} . $$ For any α 0 ∈ (0, 1), it is shown that there exists an infinite-dimensional, closed subspace of Λ X,p α0 , such that any non-identically zero function does not belong to the subspace Λ X,p α with α > α 0.