linear algebra - What does the vector space R^[0,1] mean?

While reading a book Linear Algebra Done Right, I came to knew that a vector space $\mathbf{R}^n$ represents a space with dimensions as $(x_1, x_2, ...,x_n)$, but there were other vector spaces that I could not understand.

There was a statement as Ref: 1.35

The set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbf{R}^{[0,1]}$

What kind of space does $\mathbf{R}^{[0,1]}$ represent? Is this a space that can continuously be from $0$ dimension to $1$ dimension?

Another statement, Ref: 1.35

The set of differentiable real-valued functions on $\mathbf{R}$ is a subspace of $\mathbf{R}^\mathbf{R}$

What kind of space is $\mathbf{R}^\mathbf{R}$?

Similarly, there were other subspaces as, $\mathbf{R}^{(0,\ 3)}$ and $\mathbf{R}^{(-4,\ 4)}$

Explain me how can I visualize such spaces. If you can explain with the proof too, that will be great.

Answer

When $A$ and $B$ are sets, $A^B$ represents a set. The elements of $A^B$ are functions from $B$ to $A$.

A vector of $\mathbb R^{[0,1]}$ is a function that maps $[0,1]$ to the real numbers. For example,

$$f:[0,1]\to \mathbb R\\ f(x)=x^2$$ is one such vector.

Addition and scalar multiplication in the vector space are defined as

• If $f, g\in\mathbb R^{[0,1]}$, then $f+g$ is defined as $(f+g)(x)=f(x)+g(x)$


• If $f\in\mathbb R^{[0,1]}$ and $\lambda\in\mathbb R$ then $\lambda f$ is defined as $(\lambda f)(x)=\lambda\cdot f(x)$

It's easy to see that the functions $f+g$ and $\lambda f$ above are both also elements of $\mathbb R^{[0,1]}$