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Following up on Qiaochu's query, one way of distinguishing a finite-dimensional V $V$ from an infinite one is that there exists a space W $W$ together with maps e: $e: W \otimes V \to kk$, f: $f: k \to V \otimes W W$ making the usual triangular equations hold(I hope it is clear what those equations are; cf. categorical adjunctions). The data (W, $(W, e, f) f)$ is uniquely determined up to canonical isomorphism, namely W $W$ is canonically isomorphic to the dual of V; $V$; the e $e$ is of course the evaluation pairing. (While it is hard to write down an explicit formula for f: $f: k \to V \otimes V^* V^*$ without referring to a basis, it is nevertheless independent of basis: is the same map no matter which basis you pick, and thus canonical.) By swapping V $V$ and W $W$ using the symmetry of the tensor, there are maps $V \otimes W \to kk$, $k \to W \otimes V V$ which exhibit V $V$ as the dual of W, $W$, hence V $V$ is canonically isomorphic to the dual of its dual.

Just to be a tiny bit more explicit, the inverse to the double dual embedding $V \to V^{**} V^{**}$ would be given by

V^{**}

$$V^{\ast\ast} \to V \otimes V^* \otimes V^{**V^{\ast\ast} \to V V$$

where the description of the maps uses the data above.

1

Following up on Qiaochu's query, one way of distinguishing a finite-dimensional V from an infinite one is that there exists a space W together with maps e: W \otimes V \to k, f: k \to V \otimes W making triangular equations hold (I hope it is clear what those equations are; cf. categorical adjunctions). The data (W, e, f) is uniquely determined up to canonical isomorphism, namely W is canonically isomorphic to the dual of V; the e is of course the evaluation pairing. (While it is hard to write down an explicit formula for f: k \to V \otimes V^* without referring to a basis, it is nevertheless independent of basis: is the same map no matter which basis you pick, and thus canonical.) By swapping V and W using the symmetry of the tensor, there are maps V \otimes W \to k, k \to W \otimes V which exhibit V as the dual of W, hence V is canonically isomorphic to the dual of its dual.

Just to be a tiny bit more explicit, the inverse to the double dual embedding V \to V^{**} would be given by

V^{**} \to V \otimes V^* \otimes V^{**} \to V

where the description of the maps uses the data above.