Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
The generalization of the above norms to an infinite number of components leads to , with norms
Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article.
There are also norms on spaces of matrices (with real or complex entries), the so-called matrix norms.
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.