Real analysis is a branch of math that takes a closer look at the concepts developed in introductory calculus: differentiation, integration, limits, and infinite series. If you really want to fully develop these ideas, then you might start asking yourself what structure you need on your space to understand these operations. What ends up being crucial is a sense of nearness (a topology

https://en.wikipedia.org/wiki/Topological_space), a way to measure distances (a metric

https://en.wikipedia.org/wiki/Metric_(mathematics)), a way to measure lengths (a norm

https://en.wikipedia.org/wiki/Norm_(mathematics)), and a way to measure angles (an inner product

https://en.wikipedia.org/wiki/Inner_product_space). This is all roughly speaking of course, but with all of these structures together on a (vector) space, it is possible to develop a fairly full theory of differential calculus of several dimensions on the real numbers.

Notice that in the above Venn diagram that most of the spaces are a subset of one of the others. This happens because each smaller space induces a property on the larger space. That is, if we start with an inner product space (a vector space with an inner product), then we have a “natural” way of writing a norm using that inner product. So an inner product induces a norm, a norm induces a metric, and a metric induces a topology. However, not every topological space is a metric space, not every metric space is a normed vector space, and not every normed vector space is an inner product space.