MathGloss

A Hermitian inner product ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle on a complex vector space VV is a bilinear form on VV that is antilinear in the second slot. That is,

  1. ⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩\langle u+v,w\rangle = \langle u,w\rangle +\langle v,w\rangle;
  2. ⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩\langle u,v+w\rangle = \langle u,v\rangle +\langle u,w\rangle;
  3. ⟨αu,v⟩=α⟨u,v⟩\langle \alpha u ,v\rangle = \alpha\langle u,v\rangle ;
  4. ⟨u,αv⟩=α‾⟨u,v⟩\langle u ,\alpha v\rangle = \overline\alpha\langle u,v\rangle ;
  5. ⟨u,v⟩=⟨v,u⟩‾\langle u,v\rangle = \overline{\langle v,u\rangle}
  6. ⟨u,u⟩≥0\langle u,u \rangle \geq 0
  7. ⟨u,u⟩=0\langle u,u \rangle = 0 if and only if u=0u=0.

Wikidata ID: Q77583424

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