Hence these two theorems imply the boundedness theorem and the extreme value theorem. Results that give sufficient conditions for boundedness are known as multiplier theorems. The boundedness condition for linear operators on normed spaces can be restated. Barry Mazur proved uniform boundedness for elliptic curves over the rationals. Note that compactness depends only on the topology, while boundedness depends on the metric. Every weakly convergent sequence is bounded, by the uniform boundedness principle. Borisov conjecture on boundedness of Fano varieties. Note that boundedness in nouns should not be thought of as synonymous with countability. Since formula 8, boundedness and ellipticity of the bilinear form apply to formula 20. The derived category of an abelian category has several subcategories corresponding to different boundedness conditions. The resulting axiom schema is also called the axiom schema of boundedness. Note that the two notions of boundedness coincide for locally convex spaces. The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. This boundedness of the derivatives is needed to allow for differentiation under the integral sign.