I'm studying Arzela-Ascoli (particularly as a generalization of Bolzano-Weierstrass) and am wondering how to think of functions as generalizations of points in $\mathbb{R}^k$. Specifically, I'm looking for intuition motivated by basic linear algebra ideas (bases, relations between different dimensions of the space). So if I'm building toward understanding infinite dimensional function spaces, can you first describe how we move from $\mathbb{R}^k$ to finite dimensional function spaces as an intermediate step (particularly emphasizing the role of equicontinuity)?
[Math] finite dimensional function space
functional-analysislinear algebra
Best Answer
One transitional example is to look at the vector space of real-valued functions on a set with $k$ elements, such as $\{1,2,\ldots,k\}$, with addition pointwise ($(f+g)(x)=f(x)+g(x)$) and scalar multiplication similarly. This is a $k$-dimensional space of functions. Indeed, hard to tell it apart from $\mathbb R^k$. Equicontinuity is relatively easy...