Fréchet space (idea)

In mathematics, a Fréchet space is a topological vector space which is almost a Banach space, but not quite. Fréchet spaces can arise as the limit (in a category theoretic sense, or less formally) of a sequence of Banach spaces.

Suppose q is a quasi-norm on a vector space X (over the real or complex field). This means that

• q obeys the triangle inequality, q(x + y) ≤ q(x) + q(y),
• and the definiteness property, q(x) = 0 if and only if x = 0,
• but q does not necessarily respect multiplication by scalars, the way a norm does: rather than q(αx) = |α| q(x) for any scalar α, we only require q(-x) = q(x).

Then d(x, y) = q(x - y) defines a metric on X, which makes X into a metric space. The metric d is invariant under translations: d(x + z, y + z) = d(x, y) for any x, y, z. If X is a complete metric space in the metric d induced by the quasi-norm q, we say X is a Fréchet space.

In practice Fréchet spaces are most often constructed in the following way. Suppose {p_j} is a countable family of seminorms on X. (This means that each p_j obeys the triangle inequality, and unlike a quasi-norm respects multiplication by scalars, but p_j may kill some points of X: there may be x ≠ 0 having p_j(x) = 0.) Suppose further that {p_j} separates the points of X, in the sense that for any nonzero x in X, although many of the p_j may satisfy p_j(x) = 0, there is at least one p_k with p_k(x) ≠ 0. (Thus the p_j taken all together are enough to distinguish every nonzero point of X from zero.) Then

q = ∑_j≥1 p_j / (2^j (1 + p_j))

defines a quasi-norm q on X. Then we may be able to prove that (X, q) is a complete quasi-normed vector space, thus that the seminorms p_j make X into a Fréchet space. Defining it this way shows that a Fréchet space is precisely a locally convex space which is metrizable with a complete metric.

The simplest example of a Fréchet space which is not a Banach space is C^∞(X), the space of infinitely differentiable functions on a compact smooth manifold X. C^∞(X) is the intersection of the Banach spaces C^k(X) for k ∈ N. However, C^∞(X) is not closed in C^k(X) for any finite k, so we cannot simply use any of the Banach space norms on C^k(X) to norm C^∞(X); the resulting space is not complete. If instead we call each of the C^k norms a seminorm on C^∞(X), and perform the construction above, the resulting space is complete, and thus a Fréchet space.

Banach spacers beware: the dual of a Fréchet space need not be a Fréchet space, and may be much worse. An example is C_c^∞(X), the space of compactly supported smooth functions on X, whose dual is the space D′(X) of distributions on X, which is only an LF space, that is an inductive limit of Fréchet spaces.

For examples of the use of Fréchet spaces to perform functional analytic constructions where Banach space structure is absent, you might turn to Manifolds, tensor analysis and applications by Abraham, Marsden and Ratiu, or to volumes 2, 7 and 8 of the amazing Treatise on analysis by Jean Dieudonné. For more abstract information on topological vector spaces of all kinds consult Topological vector spaces by H. H. Schaefer, in the Springer GTM series.