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 .