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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeJun 8th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded that an $n$-dimensional vector space is also a vector space with a linear isomorphism to the $n$-th tensor power of the ground field. Amy Reed <a href="https://ncatlab.org/nlab/revision/diff/finite-dimensional+vector+space/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/finite-dimensional+vector+space/7">v7</a>, <a href="https://ncatlab.org/nlab/show/finite-dimensional+vector+space">current</a>

   added that an $n$-dimensional vector space is also a vector space with a linear isomorphism to the $n$-th tensor power of the ground field.

   Amy Reed

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeJun 8th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexn-fold direct sum, not the $n$-th tensor product. any tensor product of the ground field would simply be the ground field because the ground field is the tensor unit. Amy Reed <a href="https://ncatlab.org/nlab/revision/diff/finite-dimensional+vector+space/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/finite-dimensional+vector+space/7">v7</a>, <a href="https://ncatlab.org/nlab/show/finite-dimensional+vector+space">current</a>

   n-fold direct sum, not the $n$-th tensor product. any tensor product of the ground field would simply be the ground field because the ground field is the tensor unit.

   Amy Reed

   diff, v7, current