[Verse 1] In the realm where vectors dance and play Lives a shadow world of measurement's way V-star holds the functionals that gleam Linear maps that make scalars from the dream Every vector space has its faithful twin Same dimension, but looking from within [Chorus] Dual spaces measure what you cannot see Phi takes vectors, gives you numbers free V-star equals V in size so neat Double dual brings the circle complete Tensor products multiply the space Bilinear rules keep everything in place [Verse 2] When you tensor two spaces, magic grows V cross W through symbols that compose v tensor w obeys the laws divine Linear in each spot, the rules align Add inside the tensor, split apart Scalars float around like works of art [Chorus] Dual spaces measure what you cannot see Phi takes vectors, gives you numbers free V-star equals V in size so neat Double dual brings the circle complete Tensor products multiply the space Bilinear rules keep everything in place [Bridge] Functionals are measurements disguised Taking vectors, leaving them downsized Quantum states and geometry's embrace All live in this tensor product space Dimensions multiply when worlds collide V times W with structure as your guide [Verse 3] Double dual reveals a truth profound Every vector can be safely found As a functional on the dual domain Evaluation games that break the chain Finite spaces loop back to themselves Through the magic of these abstract shelves [Chorus] Dual spaces measure what you cannot see Phi takes vectors, gives you numbers free V-star equals V in size so neat Double dual brings the circle complete Tensor products multiply the space Bilinear rules keep everything in place [Outro] From linear functionals to tensor's art Algebra reveals its beating heart Measurements and products intertwined In the dual space of the abstract mind
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