<s>
In	O
mathematics	O
,	O
the	O
category	B-Algorithm
of	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
is	O
the	O
category	O
whose	O
objects	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
whose	O
morphisms	O
are	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
between	O
them	O
.	O
</s>
<s>
This	O
is	O
a	O
category	O
because	O
the	O
composition	B-Application
of	O
two	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
is	O
again	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Fixing	O
a	O
topological	O
field	O
K	O
,	O
one	O
can	O
also	O
consider	O
the	O
subcategory	O
TVectK	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
over	O
K	O
with	O
continuous	O
K-linear	O
maps	O
as	O
the	O
morphisms	O
.	O
</s>
<s>
The	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
"	O
initial	O
object	O
"	O
or	O
"	O
initial	O
structure	O
"	O
with	O
respect	O
to	O
the	O
given	O
data	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
indiscrete	O
iff	O
it	O
is	O
the	O
initial	O
structure	O
with	O
respect	O
to	O
the	O
empty	O
family	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
discrete	O
iff	O
it	O
is	O
the	O
initial	O
structure	O
with	O
respect	O
to	O
the	O
family	O
of	O
all	O
possible	O
linear	O
maps	O
into	O
all	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
This	O
left	O
adjoint	O
defines	O
"	O
free	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
"	O
.	O
</s>
