to the case where is replaced by a general space with and replaced by Equivalent definitions Let be a linear operator between normed spaces. The first four definitions are always equivalent, and if in addition then they are all equivalent: If then the sets in the last two rows will be empty, and consequently their supremums over the set will equal instead of the correct value of If the supremum is taken over the set instead, then the supremum of the empty set is and the formulas hold for any If is bounded then and where is the transpose of which is the linear operator defined by Importantly, a linear operator is not, in general, guaranteed to achieve its norm on the closed unit ball meaning that there might not exist any vector of norm such that (if such a vector does exist and if then would necessarily have unit norm ). R.C. James proved James's theorem in 1964, which states that a Banach space is reflexive if and only if every bounded linear function achieves its norm on the closed unit ball. It follows, in particular, that every non-reflexive Banach space has some bounded linear functional (a type of bounded linear operator) that does not achieve its norm on the closed unit ball. Properties The operator norm is indeed a norm on the space of all bounded operators between and . This means The following inequality is an immediate consequence of the definition: The operator norm is also compatible with the composition, or multiplication, of operators: if , and are three normed spaces over the same base field, and and are two bounded operators, then it is a sub-multiplicative norm, that is: For bounded operators on , this implies that operator multiplication is jointly continuous. It follows from the definition that if a sequence of operators converges in operator norm, it converges uniformly on bounded sets. Table of common operator norms By choosing different norms for the domain, used in computing , and the codomain, used in computing , we obtain different values for the operator norm. Some common operator norms are easy to calculate, and others are NP-hard. Except for the NP-hard norms, all these norms can be calculated in operations (for an matrix), with the exception of the norm (which requires operations for the exact answer, or fewer if you approximate it with the power method or