finite-dimensional vector space


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Basic facts




A vector space is finite-dimensional if it admits a finite basis. The category FinVect of finite-dimensional vector spaces is of course the full subcategory of Vect whose objects are finite-dimensional.


Compact closure


A vector space VV is finite-dimensional precisely if the hom-functor hom(V,):VectSet\hom(V, -): Vect \to Set preserves filtered colimits.


Every vector space WW is the filtered colimit of the diagram of finite-dimensional subspaces WWW' \subseteq W and inclusions between them; applying this to W=VW = V, the condition that hom(V,)\hom(V,-) preserves filtered colimits implies that the canonical comparison map

colim fdVVhom(V,V)hom(V,V)colim_{fd\; V' \subseteq V} \hom(V, V') \to \hom(V, V)

is an isomorphism, so some element [f][f] in the colimit represented by f:VVf: V \to V' gets mapped to 1 V1_V, i.e., if=1 Vi \circ f = 1_V for some inclusion i:VVi: V' \hookrightarrow V. This implies ii is an isomorphism, so that VV is finite-dimensional.

In the converse direction, observe that hom(V,)\hom(V, -) has a right adjoint (and in particular preserves filtered colimits) if VV is finite-dimensional.

To see this, first notice that the dual vector space V *V^\ast of functionals f:Vkf: V \to k to the ground field is a dual object to VV in the monoidal category sense, so that there is a counit eva:V *Vkeva \colon V^\ast \otimes V \to k taking fvf(v)f \otimes v \mapsto f(v). The unit is uniquely determined from this counit and can be described using any basis e ie_i of VV and dual basis f jf_j as the map

kVV *k \to V \otimes V^\ast

taking 1 ie if i1 \mapsto \sum_i e_i \otimes f_i. We thus have an adjunction ( kV)(V *)(- \otimes_k V) \; \dashv (- \otimes V^\ast), which is mated to an adjunction hom(V,)hom(V *,)\hom(V, -) \dashv \hom(V^\ast, -) by familiar hom-tensor adjunctions; thus hom(V,)\hom(V, -) has a right adjoint.

This means that


Finite-dimensional vector spaces are exactly the compact objects of Vect in the sense of locally presentable categories, but also the compact = dualizable objects in the sense of monoidal category theory. In particular the category FinVect is a compact closed category.

Revised on November 4, 2016 07:28:49 by Urs Schreiber (