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