Projective limit of locally convex spaces - Imported Knowlpedia Content

Definition

Let $(E_i,\pi_{ij})$ be a projective system of locally convex spaces: for $i\le j$ , the map $\pi_{ij}:E_j\to E_i$ is continuous and linear, with $\pi_{ii}$ the identity and $\pi_{ik}=\pi_{ij}\pi_{jk}$ . Its projective limit is

\varprojlim_i E_i =\{(x_i)\in\prod_iE_i:\pi_{ij}(x_j)=x_i\text{ whenever }i\le j\},

equipped with the subspace topology inherited from the product's product topology. The coordinate projections are continuous linear maps and jointly determine this topology. In particular, the resulting space is locally convex.

Universal property

If $F$ is locally convex and continuous linear maps $f_i:F\to E_i$ satisfy $\pi_{ij}f_j=f_i$ , there is a unique continuous linear map $f:F\to\varprojlim_iE_i$ whose $i$ -th coordinate is $f_i$ . This makes the construction the categorical limit in locally convex spaces and characterizes it up to canonical topological isomorphism.

Seminorms and completeness

The limit topology is generated by seminorms $p\circ\operatorname{pr}_i$ , where $p$ is a continuous seminorm on $E_i$ . If all $E_i$ are Hausdorff, the compatibility equations define a closed subspace of the product. Consequently, a projective limit of complete Hausdorff locally convex spaces is complete. A countable projective limit of Fréchet spaces is again Fréchet Schaefer–Wolff, Chapter II.

Examples and cautions

For a compact smooth manifold $M$ , the inclusions $C^{k+1}(M)\to C^k(M)$ form a projective system of Banach spaces whose limit is $C^\infty(M)$ with its usual Fréchet topology. Projective limits reverse the direction of the bonding maps and should not be confused with locally convex inductive limits. If Hausdorffness is omitted, the compatible-tuple space need not be separated.

References

Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter II on products and projective limits.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967; Dover reprint, 2006. Dover publisher record. Relevant: Chapter 10 on projective limits.