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  "body": "\n## Summary\n\nA barrel is a closed, convex, balanced and absorbing set.\nA barrelled space is a [[locally convex space|B6GY2KJMWW]] in which every barrel is a neighbourhood of $0$.\nBarrelled spaces are the spaces where the uniform boundedness principle (Banach–Steinhaus theorem) holds in its usual form.\nBanach spaces, [[Fréchet spaces|3GAA614673]] and [[LF-spaces|7THJSH9SFP]] are barrelled.\n\n## Explanations\n\n### L1 human\n\nA barrel is a set that looks like a closed unit ball: it is closed, convex, symmetric, and big enough in every direction.\nIn a barrelled space every such set is automatically a genuine neighbourhood of $0$.\nThis rules out sets that are \"thin\" in infinitely many directions at once.\nThe reason one cares is that pointwise bounded families of continuous linear maps are then equicontinuous.\nIn a Banach space this is the classical uniform boundedness principle, proved with the Baire category theorem.\n",
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