A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {p_{i}}_{{i∈N}} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with p_{i}(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {p_{i}}:

d(x, y) = ∑_{i=1}^{∞} 1/2^{i} p_{i}(x – y)/(1 + p_{i}(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

- it is complete as a topological vector space;
- it is a Hausdorff space;
- its topology is induced by a countable family of seminorms {p
_{i}}_{{i∈N}}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|p_{k}(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, x_{n} → x, iff p_{i}(x_{n} – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms p_{k} on V inducing its topology and such that:

- if x ∈ V and p
_{k}(x) = 0 ∀ k ≥ 0, then x = 0 (separation property); - if (x
_{n}) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (x_{n}) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C^{∞}(U), the space of smooth functions on the open set U ⊆ R^{n} or more generally the vector space C^{∞}(M), where M is a differentiable manifold.

For each open set U ⊆ R^{n} (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||_{K,I} := sup_{x∈K} |(∂^{|I|}/∂x^{I} (ƒ)) (x)|, ƒ ∈ C^{∞}(U)

Each ||.||_{K,I} defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C^{∞}(U) into a Fréchet space. The sets of the form

|ƒ ∈ C^{∞}(U)| ||ƒ – g||_{K,I} < ε

with fixed g ∈ C^{∞}(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

p_{K,D}(ƒ) := sup_{x∈K}|D(ƒ)|

is a seminorm. The family of all the seminorms p_{K,D} with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing C_{M}^{∞}(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence C_{M}^{∞}(U) is a Fréchet space. Let indeed {V_{j}} be a countable open cover of U by open coordinate subsets, and let, for each j, {K_{j,i}} be a countable family of compact subsets of V_{j} such that ∪_{i} K_{j,i} = V_{j}. We have the countable family of seminorms

p_{K,I} := sup_{x∈K} |(∂^{|I|}/∂x^{I} (ƒ)) (x)|, K ∈ {K_{j,i}}

inducing the topology. C_{M}^{∞}(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {q_{i})_{i∈N} of seminorms satisfying

q_{i}(ƒg) ≤q_{i} (ƒ) q_{i}(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1) for each open set U ⊆ M, F(U) is a Fréchet space;

(2) for each open set U ⊆ M and for each open cover {U_{i}} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(U_{i}), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.