In Sheldon Axler's Linear Algebra Done Right third edition the following is given as an example of a subspace:
The set of differentiable real-valued functions on $\mathbb R$ is a subspace of $\mathbb R^{\mathbb R}$
I'm looking for an intuitive explanation of the statement? Letting $S$ be the set of all differentiable real-valued functions, in order for the statement to be true, $S$ must be a subset of $\mathbb R^{\mathbb R}$(a subspace needs to be a subset).
–How can $S \subset \mathbb R^{\mathbb R}$ when $S$ is a set containing functions and $\mathbb R^{\mathbb R}$ is a set containing real numbers?
–What are the elements in $\mathbb R^{\mathbb R}$? How can we think of $ \mathbb R^{\mathbb R}$ as a tuple?
Best Answer
$\mathbb R^{\mathbb R}$ denotes the set of all maps from $\mathbb R$ to $\mathbb R$.
So $S \subset \mathbb R^{\mathbb R}$ is given.
To check if $S$ is a linear subspace, see if the zero element is in $S$, see if the sum of two members of $S$ is in $S$, and similarly for multiplication by a scalar.