Basic Analysis – Gently Done Topological Vector Spaces

10: Fréchet Spaces 123
0 and q n (y ν −y) → 0 for every n ∈ N, ρ n ((x ν ,y ν )−(x,y)) → 0 for every n ∈ N,
(x ν ,y ν ) → (x,y) in the vector topology determined by the family {ρ n }.
Let d X , d Y and d X×Y be the translation invariant metrics constructed from
the appropriate families of seminorms, i.e.,
and
d X (x,x ′ ) =
∞�
n=1
1
2 n
d X×Y ((x,y),(x ′ ,y ′ )) =
p n (x−x ′ )
1+p n (x−x ′ ) , d Y (y,y′ ) =
∞�
n=1
1
2 n
∞�
n=1
1
2 n
p n (y −y ′ )
1+p n (y −y ′ ) ,
p n (x−x ′ )+q n (y −y ′ )
1+p n (x−x ′ )+q n (y −y ′ ) ,
for x,x ′ ∈ X and y,y ′ ∈ Y. We have d X (x,x ′ ) ≤ d X×Y ((x,y),(x ′ ,y ′ )) and also
d Y (y,y ′ ) ≤ d X×Y ((x,y),(x ′ ,y ′ )), so that if ((x k ,y k )) k∈N is a Cauchy sequence
in X ×Y, then its components (x k ) and (y k ) are Cauchy sequences in X and Y,
respectively, and therefore converge to x and y, say. But then p n (x k ) → p n (x) and
q n (y k ) → q n (y) as k → ∞ for each n ∈ N. This means that ((x k ,y k )) converges
to (x,y) with respect to the metric d X×Y , i.e., X ×Y is a Fréchet space.
Definition 10.25 Let T : X → Y be a linear map from the topological vector
space X into the topological vector space Y. The graph of T is the subset Γ(T)
of X ×Y given by
Γ(T) = {(x,Tx) : x ∈ X}.
Theorem 10.26 (Closed Graph theorem) Suppose that X and Y are Fréchet
spaces and T : X → Y is a linear map from X into Y. Then T is continuous if
and only if it has a closed graph in X ×Y.
Proof Suppose that T is continuous and suppose that (xn ,Txn ) → (x,y) in
X×Y. Then xn → x and so Txn → Tx. But Txn → y and it follows that y = Tx
and therefore (x,y) = (x,Tx) ∈ Γ(T), that is, Γ(T) is closed.
Conversely, suppose that Γ(T) is closed in X×Y. Then Γ(T) is a closed linear
subspace of the Fréchet space X×Y and so is itself a Fréchet space—with respect
to the restriction of the metric dx×Y . Let πX : X ×Y → X and πY : X ×Y → Y
be the projection maps. By definition of the topologies on these spaces, it is clear
that both πX and πY are continuous. Moreover, πX : X ×X is one-one and onto
X and so, by the inverse mapping theorem, Corollary 10.22, its inverse, π −1
X is
continuous from X into X ×Y. But then T is given by
T : x π−1
X
↦−→ (x,Tx) πY
↦−→ Tx,
i.e., T = πY ◦π −1
X , the composition of two continuous maps, and so is continuous.