A topological vector space
over
is a vector space over the field
that has a topology with which scalar multiplication and addition are continuous mappings.

In the following, for all topological vector spaces, we shall use the
Hausdorff property be assumed.
Let
be a topological space with a topology
as a system of open sets
and
, then denote
the set of all neighbourhoods from the point
,
the set of all open Neighbourhoods from the point
,
the set of all closed neighbourhoods of point
.
If no misunderstanding about the underlying topological
space can occur, the index
is not included as a designation of the topology used.
In convergence statements in the real numbers one usually considers only
neighbourhood. In doing so, one would actually have to consider in topological spaces for arbitrary neighbourhoods from
find an index bound
of a net
above which all
lie with
. However, since the
neighbourhoods are an neighbourhood basis, by the convergence definition one only needs to show the property for all neighbourhoods with
.
Let
be a topological space,
,
an index set (partial order) and
a mesh. The convergence of
against
is then defined as follows:
.
(where "
" for
is the partial order on the index set).
Let
be a topological space,
and
the set of all neighbourhoods of
.
is called the neighbourhood basis of
if for every :
.
Let
be a normed space, then the
spheres form

an ambient basis of
the set of all environments of
of
.
Let
be a toplogic space with chaotic topology
.
- Determine
for any
.
- Show that any sequence
converges in
against any limit
.
Let
be a metric space with the discrete topology given by the metric:
.
- Determine
for any
.
- How many sets make up
minimal for any
?
- Formally state all sequences
in
that converge to a limit
!
Let
be a topological space and
be the system of open sets, that is:
.
Let
be a topological space on the basic set of real numbers. However, the topology does not correspond to the Euclic topology over the set
, but the open sets are defined as follows.

- Show that
is a topological space.
- Show that the sequence
does not converge to
in the topological space
.
Here
is the complement of
in
.
By the system of open sets in a topology
the closed sets of the topology are also defined at the same time as their complements.
Let
be a topological space and
be the system of open sets.

Let
be a topological space and
, then the open
kernel
of
is the union of all open subsets of
.
.
Let
be a topological space. The closed hull
of
is the intersection over all closed subsets of
containing
and
is open.

The topological edge
of
is defined as follows:

In metric spaces, one can still work with the natural numbers as countable index sets. In arbitrary topological spaces one has to generalize the notion of sequences to the notion of nets.
Let
be a topological space and
an index set (with partial order), then
denotes the set of all families indexed by
in
:

Let
be a vector space, then
denotes the set of all finite sequences with elements in
:

An algebra
over the field
is a vector space over
in which a multiplication is an inner join

is defined where for all
and
the following properties are satisfied:

A topological algebra
over the field
is a topological vector space
over
, where also multiplication is

is a continuous inner knotting.
Continuity of multiplication means here:

The topology is called multiplicative if holds:

In describing topology, the Topologization Lemma for Algebras shows that the topology can also be described by a system of Gaugefunctionals
The algebra
is called unital if it has a neutral element
of multiplication. In particular, one defines
for all
. The set of all invertible (regular) elements is denoted by
. Non-invertible elements are called singular.
Consider the set
of square
matrices with matrix multiplication and the maxmum norm of the components of the matrix. Try to prove individual properties of an algebra (
is a non-commutative unitary algebra). For the proof that
with matrix multiplication is also a topological algebra, see Topologization Lemma for Algebras.
Let
be a topological algebra over the field
,
and
be subsets of
, then define

Draw the following set
of vectors as sets of points in the Cartesian coordinate system
with
and
and the following intervals
:
.
.
.
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