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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nAn *inner product space* over $\\mathbb{K}$ is a vector space $V$ over $\\mathbb{K}$ together with an [[inner product|0NB0AWGJBW]] $\\langle \\cdot, \\cdot \\rangle$ on $V$.\n\n## Remarks\n\nA finite-dimensional real inner product space is a *Euclidean space*.\nEach inner product space has the [[induced norm|0MFJJV97RJ]] $\\norm{x} = \\sqrt{\\langle x, x \\rangle}$.\nAn inner product space that is complete for this norm is a *Hilbert space*.\nEach finite-dimensional inner product space is complete.\n\n## Explanations\n\n### L1 human\n\nAn inner product space is a vector space where we can measure lengths and angles.\nThe space $\\R^n$ with the dot product is the main example.\nMuch of plane geometry, such as the Pythagorean theorem, works in every inner product space.\n",
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