The finest locally convex topological vector space (TVS) topology on 
the tensor product of two locally convex TVSs, making the canonical map 
(defined by sending 
to 
) separately continuous is called the inductive topology or the 
-topology. When 
is endowed with this topology then it is denoted by 
and called the inductive tensor product of 
and 
Preliminaries
Throughout let 
and 
be locally convex topological vector spaces and 
be a linear map.
- is a topological homomorphism or homomorphism, if it is linear, continuous, and
is an open map, where 
the image of 
has the subspace topology induced by
- If
is a subspace of then both the quotient map 
and the canonical injection 
are homomorphisms. In particular, any linear map can be canonically decomposed as follows: 
where 
defines a bijection.
- The set of continuous linear maps
(resp. continuous bilinear maps 
) will be denoted by 
(resp. 
) where if is the scalar field then we may instead write 
(resp. 
).
- We will denote the continuous dual space of by
and the algebraic dual space (which is the vector space of all linear functionals on 
whether continuous or not) by 
- To increase the clarity of the exposition, we use the common convention of writing elements of with a prime following the symbol (e.g.
denotes an element of and not, say, a derivative and the variables 
and need not be related in any way).
- A linear map
from a Hilbert space into itself is called positive if 
for every 
In this case, there is a unique positive map 
called the square-root of such that 
- If
is any continuous linear map between Hilbert spaces, then 
is always positive. Now let 
denote its positive square-root, which is called the absolute value of 
Define 
first on 
by setting 
for 
and extending 
continuously to 
and then define on 
by setting 
for 
and extend this map linearly to all of 
The map 
is a surjective isometry and 
- A linear map
is called compact or completely continuous if there is a neighborhood of the origin in such that 
is precompact in
- In a Hilbert space, positive compact linear operators, say have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:
- There is a sequence of positive numbers, decreasing and either finite or else converging to 0,
and a sequence of nonzero finite dimensional subspaces 
of 
(
) with the following properties: (1) the subspaces are pairwise orthogonal; (2) for every 
and every 
; and (3) the orthogonal of the subspace spanned by 
is equal to the kernel of
Notation for topologies
Universal property
Suppose that is a locally convex space and that 
is the canonical map from the space of all bilinear mappings of the form 
going into the space of all linear mappings of 
Then when the domain of is restricted to 
(the space of separately continuous bilinear maps) then the range of this restriction is the space 
of continuous linear operators 
In particular, the continuous dual space of is canonically isomorphic to the space 
the space of separately continuous bilinear forms on 
If 
is a locally convex TVS topology on ( with this topology will be denoted by 
), then is equal to the inductive tensor product topology if and only if it has the following property:
- For every locally convex TVS
if is the canonical map from the space of all bilinear mappings of the form going into the space of all linear mappings of 
then when the domain of is restricted to (space of separately continuous bilinear maps) then the range of this restriction is the space 
of continuous linear operators 
See also
denotes the coarsest topology on
or
denotes
denotes weak-* topology on
or
denotes
induces a map
defined by
denotes the topology of bounded convergence on
or
denotes
denotes the topology of bounded convergence on
or
denotes
