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  "body": "\n## Definition\n\nLet $H$ be an [[inner product space|0QVNCJXRY3]] over $\\R$ or $\\C$ with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $\\norm{x} = \\sqrt{\\ip{x}{x}}$ be the [[induced norm|0MFJJV97RJ]] and $d(x, y) = \\norm{x - y}$ the induced metric.\nThe space $H$ is a *Hilbert space* if the metric space $(H, d)$ is [[complete|0BY52JW6NK]].\n\n## Remarks\n\nExample: $\\ell^2$, the space of real sequences $x$ with $\\sum_n x_n^2 \u003c \\infty$ and $\\ip{x}{y} = \\sum_n x_n y_n$.\nThe series converges by the Cauchy–Schwarz inequality.\nIt is complete: a Cauchy sequence converges in each coordinate, and the coordinatewise limit is in $\\ell^2$ and is the limit in norm.\n\nNon-example: $C([0, 1])$ with $\\ip{f}{g} = \\int_0^1 f g \\, dx$ is an inner product space but not complete.\nThe continuous functions $f_k$ equal to $0$ on $[0, 1/2]$, to $1$ on $[1/2 + 1/k, 1]$ and linear in between form a Cauchy sequence.\nTheir only possible limit is the indicator function of $(1/2, 1]$, which is not continuous.\n\nEvery finite-dimensional inner product space is a Hilbert space.\nAn inner product space that is not complete is often called a pre-Hilbert space, and its completion is a Hilbert space.\n",
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