Tensor products

• Direct Sum vs. Direct Product vs. Tensor Product
  A direct sum restricts infinite tuples to finite support, a direct product allows all tuples, and a tensor product combines spaces bilinearly.
• How to Prove an Element Is Nonzero in a Tensor Product
  Over a field, detect a nonzero tensor by applying linear functionals; over a ring, use a balanced map or compute the module relations.
• Is the Tensor Product of a Module Element and Zero Always Zero?
  In a tensor product of modules, m ⊗ 0 and 0 ⊗ n are always zero. Bilinearity proves this, but sums of pure tensors need separate analysis.
• Tensor Product vs. Cartesian Product in Quantum Systems
  For two quantum subsystems, the joint Hilbert space is a tensor product, not a Cartesian product of state vectors; see dimensions, product states, and entanglement.
• Tensor Product vs. Direct Product for Three Spin-1/2 Particles
  For three spin-1/2 particles, the composite Hilbert space is an eight-dimensional tensor product; total-spin sectors give a direct-sum decomposition.
• How to Test Whether an Element of a Tensor Product Is Zero
  To test whether a tensor is zero, use a module presentation or a separating map; pure tensors and vector-space tensors have useful special criteria.
• What Is the Tensor Product of a Vector with Itself?
  The tensor product of a vector with itself is an element of V⊗V; in coordinates, it is the outer product with entries v_i v_j.
• When Is an Element of a Tensor Product Zero?
  For vector spaces over a field, a pure tensor is zero exactly when one factor is zero; sums and tensors over general rings need separate tests.
• When Is the Tensor Product Commutative?
  Tensor products of modules over a commutative ring are symmetric up to a natural flip isomorphism, but the factors are not literally interchangeable.