**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
.