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  "body": "\n## Definition\n\nLet $V$ be a vector space over $\\C$.\nA *sesquilinear form* on $V$ is a map $\\varphi \\colon V \\times V \\to \\C$ such that, for all $u, v, w \\in V$ and $a \\in \\C$:\n\n1. $\\varphi(u + v, w) = \\varphi(u, w) + \\varphi(v, w)$ and $\\varphi(au, w) = a\\, \\varphi(u, w)$.\n2. $\\varphi(u, v + w) = \\varphi(u, v) + \\varphi(u, w)$ and $\\varphi(u, aw) = \\bar a\\, \\varphi(u, w)$.\n\n## Remarks\n\nHere the form is linear in the first argument; many authors, especially in physics, use linearity in the second argument instead.\nA sesquilinear form is *Hermitian* when $\\varphi(v, u) = \\overline{\\varphi(u, v)}$ for all $u, v$.\nOn $\\C^n$, each sesquilinear form is $\\varphi(x, y) = y^* M x$ for one $n \\times n$ complex matrix $M$.\nA complex [[inner product|0NB0AWGJBW]] is a positive definite Hermitian sesquilinear form.\n\n## Explanations\n\n### L1 human\n\nOver the complex numbers, bilinear forms cannot be positive, because $\\varphi(iv, iv) = -\\varphi(v, v)$.\nSesquilinear forms fix this by conjugating the scalar in one argument.\nThe standard inner product $\\sum_i x_i \\overline{y_i}$ on $\\C^n$ is the main example.\n",
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