It is simple to prove that this is the same condition on *L* as continuity, for the topologies induced from the norms.

In the case of a matrix *A* acting as a linear transformation, from **R**^{m} to **R**^{n}, or from **C**^{m} to **C**^{n}, one can prove directly that *A* must be bounded. In fact the function

*f*(*v*) = ||*A*(*v*)||

In general the **operator norm** of a *bounded* linear transformation *L* from *V* to *W*, where *V* and *W* are both normed real vector spaces (or both normed complex) vector spaces is defined as the *supremum* of the ||*L*(*v*)|| taken over all *v* in *V* of norm 1. This definition uses the property ||*c.v*|| = |*c*|.||*v*|| where *c* is a scalar, to restrict attention to *v* with ||*v*|| = 1. Geometrically we need (for real scalars) to look at one vector only on each *ray* out from the origin 0.

Note that there are two different norms here: that in *V* and that in *W*. Even if *V* = *W* we might wish to take distinct norms on it. In fact given two norms ||.|| and |||.||| on *V*, the identity operator on *V* will have an operator norm, in passing from *V* with ||.|| as norm to *V* with |||.|||, only if we can say

- |||
*v*||| <*C*.||*v*||

This is, however, a phenomenon of finite dimensions: that all norms will turn out to be equivalent: they stay within constant multiples of each other, and from a topological point of view they give the same open sets. This all fails for infinite-dimensional spaces. This can be seen, for example, by considering the differentiation operator D, as applied to trigonometric polynomials. We can take the root mean square as norm: since D(e^{inx}) = *in*e^{inx}, the norms of D applied to the finite-dimensional subspaces of the Hilbert space grow beyond any bounds. Therefore an operator as fundamental as D can fail to have an operator norm.

A basic theorem applies the Baire category theorem to show that if *L* has as domain and range Banach spaces, it will be bounded. That is, in the example just given, D must not be defined on all square-integrable Fourier series; and indeed we know that they can represent continuous but nowhere differentiable functions. The intuition is that if *L* magnifies norms of some vectors by as large a number as we choose, we should be able to *condense singularities* - choose a vector *v* that *sums up* others for which it would be contradictory for ||*L*(*v*)|| to be finite - showing that the domain of *L* cannot be the whole of *V*.