nLab
linear map

Contents

Definition

Discrete

A KK-linear map (also KK-linear function, KK-linear operator, or KK-linear transformation) is a morphism in KK-Vect (or KK-Mod), that is a homomorphism of vector spaces (or modules). Often one suppresses mention of the field (or commutative ring or rig) KK.

In elementary terms, a KK-linear map between KK-linear spaces VV and WW is a function T:VWT\colon V \to W such that

T(rx+y)=rT(x)+T(y), T(r x + y) = r T(x) + T(y) ,

for xx and yy elements of VV and rr an element of KK. (It is an easy exercise that this one identity is enough to ensure that TT preserves all linear combinations.)

Topological

The morphisms between topological vector spaces are of course the continuous linear maps. Between Banach spaces (including of course Hilbert spaces), these are the same as the bounded linear maps, so they're often called bounded operators (with linearity tacitly assumed).

In this context, linear operators are more general; they are (in general) only partial functions. However, we still require the domain of the partial function to be a linear subspace, after which the definition above applies. Because one typically restricts attention to complete spaces, the densely-defined operator?s (where the domain is a dense subspace) are the most general needed. To specify that the domain of a linear operator T:VWT\colon V \to W is all of VV, one may use a non-‘operator’ term, such as linear mapping.

Notice that we do not require partially-defined linear operators to be continuous; see unbounded operator. However, we have the theorem that any densely-defined, continuous, linear map T:VWT\colon V \to W, with WW complete and Hausdorff, extends uniquely to all of VV. Thus one typically assumes that a continuous (or bounded) linear operator TT is defined on all of VV while an arbitrary linear operator TT is defined only on a dense subspace of VV.

Terminology

In elementary mathematics (at least as taught in the United States, perhaps elsewhere?), the term ‘linear function’ is usually used more generally, for an affine map (but still between vector spaces); in this same context, the term ‘linear transformation’ is often used instead for specifically linear maps. (Another difference at this level is that ‘linear functions’ are usually scalar-valued, while ‘linear transformations’ are usually vector-valued.)

In operator theory?, one sometimes distinguishes ‘linear maps’ (defined everywhere, but not necessarily continuous in general) from ‘linear operators’ (partially defined in general, but assumed to be defined everywhere if continuous between complete Hausdorff spaces). There is also a tendency for ‘operator’ to be used only for (possibly partial) endomorphisms, that is T:VVT\colon V \to V; then operators may be composed, giving rise to an operator algebra. If VV is a function space, then an endomorphism of VV is an operator in the sense of higher-order logic; the more general meaning of ‘linear operator’ is abstracted from this.

Revised on February 28, 2014 14:21:46 by Urs Schreiber (89.204.139.212)