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  "body": "\n## Definition\n\nLet $K$ be a complex vector space and let $[\\cdot, \\cdot]$ be a [[Hermitian form|48P4A1CX85]] on $K$.\nThe pair $(K, [\\cdot, \\cdot])$ is a *Krein space* if there are subspaces $K_+$ and $K_-$ of $K$ such that:\n\n1. $K = K_+ \\oplus K_-$ as a direct sum of vector spaces.\n2. $[x, y] = 0$ for all $x \\in K_+$ and $y \\in K_-$.\n3. $(K_+, [\\cdot, \\cdot])$ is a [[Hilbert space|HM2HHE3J51]].\n4. $(K_-, -[\\cdot, \\cdot])$ is a Hilbert space.\n\nSuch a splitting is a *fundamental decomposition* of $K$.\n\n## Remarks\n\nLet $P_\\pm$ be the projections of $K$ onto $K_\\pm$ along the decomposition, and let $J = P_+ - P_-$.\nThen $\\ip{x}{y}_J = [Jx, y]$ is a Hilbert space inner product on $K$, and $[x, y] = \\ip{Jx}{y}_J$.\nThe map $J$ is the *fundamental symmetry*; the decomposition is not unique, but all the norms $\\ip{x}{x}_J^{1/2}$ are equivalent.\nIf $K_-$ has finite dimension $\\kappa$, the space is a *Pontryagin space* $\\Pi_\\kappa$.\n\nExample: $K = \\C^2$ with $[x, y] = x_1 \\overline{y_1} - x_2 \\overline{y_2}$, $K_+ = \\C e_1$ and $K_- = \\C e_2$.\nClause 2 holds since $[e_1, e_2] = 0$, and $[x, x] = \\abs{x_1}^2$ on $K_+$ and $-[x, x] = \\abs{x_2}^2$ on $K_-$ are positive definite.\nThe vector $e_1 + e_2$ satisfies $[e_1 + e_2, e_1 + e_2] = 0$, so the form is indefinite.\n\nNon-example: $K = \\C^2$ with $[x, y] = x_1 \\overline{y_1}$.\nThe vector $e_2$ satisfies $[e_2, y] = 0$ for every $y$.\nIn a Krein space, write $e_2 = x_+ + x_-$; then $[x_+, x_+] = [e_2, x_+] = 0$ forces $x_+ = 0$, and in the same way $x_- = 0$.\nSo the form of a Krein space is nondegenerate, and this degenerate form admits no fundamental decomposition.\n",
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