Natural topologies on function spaces

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 Product topologies of locally convex vectorspaces are locally convex, from the construction of the product. The construction of the limit as the diagonal in the product, with the subspace topology, shows that it is locally convex. In particular, countable limits of Banach spaces are locally convex, hence, are Fréchet. All spaces of practical interest are locally convex for simple reasons, so demonstrating local convexity is rarely interesting. [2.0.4] Theorem: d dx : C ∞[a, b]→ C∞[a, b] is continuous. Proof: In fact, the differentiation operator is characterized via the expression of C∞[a, b] as a limit. We already know that differentiation d/dx gives a continuous map C[a, b] → Ck−1[a, b]. Differentiation is compatible with the inclusions among the C[a, b]. Thus, we have a commutative diagram C∞[a, b] )) ** . . . C[a, b] // Ck−1[a, b] // . . . C∞[a, b] 55 55 . . . C[a, b] // d dx 99 r r r r r r r r r r Ck−1[a, b] // d dx :: v v v v v v v v v v . . . Composing the projections with d/dx gives (dashed) induced maps from C∞[a, b] to the limitands, inducing a unique (dotted) continuous linear map to the limit, as in C∞[a, b] )) ** . . . C[a, b] // Ck−1[a, b] // . . . C∞[a, b] 55 k k k k k k k k 33 g g g g g g g g g g g g g g 22 e e e e e e e e e e e e e e e e e e e e e d dx OO 55 55 . . . C[a, b] // 99 r r r r r r r r r r Ck−1[a, b] // :: v v v v v v v v v v . . .