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  "body": "\n## Definition\n\nLet $V$ and $W$ be normed spaces and let $T \\colon V \\to W$ be a [[continuous linear map|0S3SMRH0Z5]].\nThe operator norm of $T$ is\n$$\\|T\\| = \\sup \\{ \\|Tx\\|_W : x \\in V, \\ \\|x\\|_V \\le 1 \\}.$$\n\n## Remarks\n\nThe [[supremum|0BTMQ04556]] is finite because $T$ is [[bounded|0F3K4C7XYW]]. If $V \\neq \\{0\\}$, then $\\|T\\| = \\sup_{x \\neq 0} \\|Tx\\|_W / \\|x\\|_V$.\nIt is the smallest $C \\ge 0$ with $\\|Tx\\|_W \\le C \\|x\\|_V$ for all $x \\in V$.\nThe operator norm is a norm on the space $L(V, W)$ of continuous linear maps. It satisfies $\\|ST\\| \\le \\|S\\| \\, \\|T\\|$. If $W$ is a [[Banach space|0BDTTM24DP]], so is $L(V, W)$.\nFor a diagonal matrix on $\\mathbb{R}^n$ with the Euclidean norm, the operator norm is the largest absolute value of a diagonal entry.\n\n## Explanations\n\n### L1 human\n\nThe operator norm tells you how much $T$ can stretch a vector at most.\nFor example, the map $x \\mapsto 3x$ on $\\mathbb{R}$ has operator norm $3$.\n",
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