• Corpus ID: 18229032

Topological vector spaces

@inproceedings{Garrett2016TopologicalVS,
  title={Topological vector spaces},
  author={Paul B. Garrett},
  year={2016}
}

• Paul B. Garrett
• Published 2016
• Mathematics

ing the above, for a (not necessarily countable) family . . . φ2 // B1 φ1 // Bo of Banach spaces with continuous linear transition maps as indicated, not recessarily requiring the continuous linear maps to be injective (or surjective), a (projective) limit limiBi is a topological vector space with continuous linear maps limiBi → Bj such that, for every compatible family of continuous linear maps Z → Bi there is unique continuous linear Z → limiBi fitting into limiBi !! . . . φ2 // B1 φ1 // Bo Z == | | | | 66 m m m m m m m m cc The same uniqueness proof as above shows that there is at most one topological vector space limiBi. For existence by construction, the earlier argument needs only minor adjustment. The conclusion of complete metrizability would hold when the family is countable. Before declaring C∞[a, b] to be a Fréchet space, we must certify that it is locally convex, in the sense that every point has a local basis of convex opens. Normed spaces are immediately locally convex, because open balls are convex: for 0 ≤ t ≤ 1 and x, y in the ε-ball at 0 in a normed space, |tx+ (1− t)y| ≤ |tx|+ |(1− t)y| ≤ t|x|+ (1− t)|y| < t · ε+ (1− t) · ε = ε