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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *normed vector space* over $\\mathbb{K}$ is a vector space $V$ over $\\mathbb{K}$ together with a [[norm|0V2WP9B1MF]] $\\norm{\\cdot}$ on $V$.\n\n## Remarks\n\nA normed space is a metric space with the distance $d(x, y) = \\norm{x - y}$.\nA normed space that is complete for this metric is a *Banach space*.\nEach [[inner product space|0QVNCJXRY3]] is a normed space with the induced norm.\nOn a finite-dimensional space, all norms are equivalent, so they define the same topology.\n\n## Explanations\n\n### L1 human\n\nA normed space is a vector space where each vector has a length.\nWith lengths, we also have distances, and so limits and continuity.\nThis is the setting of much of analysis.\n",
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