**Problem 4.** Suppose
. Show
that the set of continuous real-valued functions
on the interval
such that
is a
subspace of
if
and only if
.

*Solution.* Let
denote the set of continuous, real-valued functions
on
with the property
that
.
Then
, so
.
Conversely, suppose that
.
Given
, we have
so
. We have
,
so
. Finally, if
and
,
one has
so
. So
is a subspace of
, as
required.

**Problem 7.** Prove or give a counterexample: If
is a nonempty subset
of
such that
is closed under
addition and under taking additive inverses (meaning
whenever
), then
is a subspace of
.

*Proof.* Let
, which
is a nonempty subset of
. Then, as
is closed
under additive inverse, if
, then
; but
,
so
is closed under
taking inverses. However, notice that
is not closed under
scalar multiplication. Indeed, we have
and
,
but
.
So
is not a subspace of
.