Classical Banach spaces
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Dual space |
Reflexive |
weakly sequentially complete |
Norm |
Notes
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![{\displaystyle \mathbb {F} ^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6abc33ef38d7d1214109b904b767b7621c100d2c) |
Yes |
Yes
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Euclidean space
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![{\displaystyle \ell _{q}^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8ad5b08d8102d81587e7046ca2ccdc8b9a19ebac) |
Yes |
Yes
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![{\displaystyle \ell _{1}^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8fd039e7e3c16763960ba3ce4d947612550c9168) |
Yes |
Yes
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![{\displaystyle \ell ^{q}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/174b9ba5de2319a7cca1be35d6262fb300355386) |
Yes |
Yes
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![{\displaystyle \ell ^{\infty }}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4) |
No |
Yes
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![{\displaystyle \operatorname {ba} }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/51014839be558bf25444d1ee41fed6ed409a3bdf) |
No |
No
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![{\displaystyle \ell ^{1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1) |
No |
No
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![{\displaystyle \ell ^{1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1) |
No |
No
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Isomorphic but not isometric to
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![{\displaystyle \ell ^{\infty }}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4) |
No |
Yes
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Isometrically isomorphic to
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![{\displaystyle \ell ^{\infty }}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4) |
No |
Yes
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Isometrically isomorphic to
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![{\displaystyle \operatorname {ba} }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/51014839be558bf25444d1ee41fed6ed409a3bdf) |
No |
No
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Isometrically isomorphic to
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![{\displaystyle \ell ^{1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1) |
No |
No
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Isometrically isomorphic to
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![{\displaystyle \operatorname {ba} (\Xi )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3d6f62699155369f1af68aff00221d4ffc097a55) |
No |
No
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![{\displaystyle \operatorname {rca} (K)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6935230b898d2d6aba19e862499cd26769dea0fa) |
No |
No
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? |
No |
Yes
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? |
No |
Yes
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A closed subspace of
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? |
No |
Yes
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A closed subspace of
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![{\displaystyle L^{q}(\mu )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9568c4262b3e1c017b6f80111b7e5b3d1a0e485c) |
Yes |
Yes
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![{\displaystyle L^{\infty }(\mu )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5b7867eb72dc22e91568af1af857fd364f42458c) |
No |
Yes
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The dual is if is -finite.
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? |
No |
Yes
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is the total variation of
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? |
No |
Yes
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consists of functions such that
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![{\displaystyle \mathbb {F} +L^{\infty }([a,b])}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3ef63fd9a8ef0c7df601ba2aa141815ea86073da) |
No |
Yes
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Isomorphic to the Sobolev space
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![{\displaystyle \operatorname {rca} ([a,b])}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b8788ca02e303b567e9d47a44b0fd48a574ddbfb) |
No |
No
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Isomorphic to essentially by Taylor's theorem.
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