Basic Analysis – Gently Done Topological Vector Spaces

122 Basic Analysis
Corollary 10.23 Suppose that X and Y are Banach spaces and that T : X → Y is
a one-one continuous linear mapfrom X onto Y. Then there arepositiveconstants
a and b such that
a�x� ≤ �T(x)� ≤ b�x�
for all x ∈ X.
Proof By the corollary, S = T −1 is continuous, and so there is some C > 0 such
�S(y)� ≤ C�y� for all y ∈ Y. Replacing y by T(x) and setting a = C −1 , it follows
that
a�x� = a�S(y)� ≤ �y� = �T(x)�, for x ∈ X.
The continuity of T implies that �T(x)� ≤ b�x� for some positive constant b and
all x ∈ X.
Corollary 10.24 Suppose that T 1 ⊆ T 2 are vector topologies on a vector space
X, such that X is a Fréchet space with respect to both. Then T 1 = T 2 .
Proof The identity map from (X,T 2 ) → (X,T 1 ) is a one-one continuous linear
map. By the open mapping theorem, it is open, and therefore every T 2 open set
is also T 1 open, i.e., T 2 ⊆ T 1 , and so equality holds.
If X and Y are vector spaces over K (either both over R or both over C) then
the Cartesian product X × Y is a vector space when equipped with the obvious
component-wise linear operations, namely t(x,y) = (tx,ty) and (x,y)+(x ′ ,y ′ ) =
(x+x ′ ,y+y ′ ) for any t ∈ K, x,x ′ ∈ X and y,y ′ ∈ Y. If X and Y are topological
vector spaces, then X ×Y can be equipped with the product topology. It is not
difficult to see that this is a vector topology thus making X×Y into a topological
) are nets in X × Y converging to
vector space. Indeed, if (xν ,yν ) and (x ′ ν ,y′ ν
(x,y) and (x ′ ,y ′ ), respectively, then xν → x and x ′ ν → x′ in X and yν → y and
y ′ ν → y′ in Y. It follows that xν +x ′ ν → x+x′ and yν +y ′ ν → y+y′ and therefore
(xν ,yν ) + (x ′ ν ,y′ ν ) → (x + x′ ,y + y ′ ) in X × Y. Thus addition is continuous in
X ×Y. In a similar way, one sees that scalar multiplication is continuous.
Now, if X and Y are Fréchet spaces, then so is X × Y. In fact, if {pn } and
{qn } are countable families of determining seminorms for X and Y, respectively,
then {ρn } is a determining family for the product topology on X ×Y, where ρn is given by
ρn ((x,y)) = pn (x)+q n (y)
for n ∈ N and (x,y) ∈ X ×Y. The fact that this family does indeed determine
the product topology on X × Y follows from the equivalence of the following
statements; (x ν ,y ν ) → (x,y)inX×Y, x ν → xinX andy ν → y inY, p n (x ν −x) →
Department of Mathematics King’s College, London