Proof
We introduce a mapping
with the following
properties:
thus, for each
the
equals
at
but does not equal
at
.
We proceed to prove existence and uniqueness of such
.
Let
be an orthonormal basis of
and
for some sequence
.
For every fixed
and
we need
to
satisfy
Thus, for each fixed
we have
equations for
unknowns
.
To complete the proof of existence of
it remains to show that the homogenous system of
equations
has only the trivial solution. Let
then, by
,
for some
and, by
,
but
also
and both the terms
and
do not change sign on
.
Thus, we must have
on
.
Existence of
has been established.

It follows from such result that if
is a
-polynomial
of degree
then
would approximate it exactly.

According to the proposition
(
Integral form of Taylor
decomposition
), for a function
we
have
where
.
By polynomial approximation up to degree
,
we must
have
Indeed, if one of the derivatives
is not zero then we cannot have the polynomial approximation with the same
formula.
Thus
where
or
Consequently
We use the formula (
Cauchy
inequality
),
Then, by orthogonality of
,
and by properties of the operator
listed in the condition (
Generic
parabolic PDE
setup
),

For
being a solution of the problem
(
Generic parabolic PDE problem
),
be a solution of the problem
(
Discontinuous Galerkin
time-discretization
) and
being the polynomial approximation defined in
we decompose the error
We already have the desired estimate for
.
It remains to estimate
.
Note that both
and
are solutions of the equation of the problem
(
Discontinuous Galerkin
time-discretization
), hence, for
we
have
Consequently
We calculate the
part
By definition of
there is orthogonality of
to
-variable
polynomials up to degree
.
Thus, we perform integration by
parts.
Now the integral is zero by the orthogonality because
thus
is a
-degree
-variable
polynomial.
The
is zero at
by definition of
.

Therefore, we arrived
to
We set
and evaluate the
part
We use the formula (
Cauchy
inequality
).
We arrived
to
or
Hence
Thus
Note that
by definition of the components and
.
and we use the estimate