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DARM.  Note 14

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| If E is a Banach space, then the diagonal !D! in E x E
| is a closed subspace and splits:  Either factor E x 0
| or 0 x E is a closed complement.  Consequently, the
| diagonal is a closed submanifold of E x E.  If X
| is any manifold of class C^p, p &gt;= 1, then the
| diagonal is therefore also a submanifold.
| (I