On translation and dilation invariant seminorms

We consider complete seminormed spaces of functions of one real argument, such that the kernel of the seminorm is finite-dimensional. If the seminorm is invariant with respect to affine change of argument, we say that the space is "interesting". We proved that the maximal "interesting" space embedded into L_{1,loc}(R) is equivalent to BMO, and the maximal "interesting" space embedded into D'(R) is equivalent to the real Bloch space. Also we construct minimal "interesting" space which contains space of smooth functions with compact support.