This turns the dual into a locally convex topological vector space.

Let E be a locally convex topological vector space.

Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces.

Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets.

The Gâteaux derivative extends the concept to locally convex topological vector spaces.

A locally convex topological vector space V has a topology that is defined by some family of seminorms.

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

Locally convex topological vector spaces: here each point has a local base consisting of convex sets.

In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

Montel spaces are reflexive locally convex topological vector spaces.

The Krein-Milman theorem is stated for locally convex topological vector spaces.

The topologies of locally convex topological vector spaces A and B are given by families of seminorms.

First, when 'V' is not Locally convex topological vector space, the continuous dual may be equal to and the map Ψ trivial.

In locally convex topological vector spaces the topology τ of the space can be specified by a family P of semi-norms.

The Gâteaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.

Let X and Y be locally convex topological vector spaces, and U X an open set.

More generally, for V a locally convex topological vector space, the Choquet-Bishop-de Leeuw theorem gives the same formal statement.

With this definition, D(U) becomes a complete locally convex topological vector space satisfying the Heine-Borel property .

A locally convex topological vector space is a topological vector space in which the origin has a local base of absolutely convex absorbent sets.

Definition 5: A nuclear space is a locally convex topological vector space such that any continuous linear map to a Banach space is nuclear.

Any action of the group by continuous affine transformations on a compact convex subset of a (separable) locally convex topological vector space has a fixed point.

Formally, let be a locally convex topological vector space (assumed to be Hausdorff), and let be a compact convex subset of .

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

In terms of the open sets, a locally convex topological vector space is seminormable if and only if 0 has a bounded set (topological vector space) neighborhood.

If we are willing to use the concept of a nuclear operator from an arbitrary locally convex topological vector space to a Banach space, we can give shorter definitions as follows: