Introduction to Functional Analysis

This target of this project is to introduce some basic definition and theorem to the readers. Due to the shortage of the author, this notebook

Chapter 1 Metric Space

This chapter is to introduce metric, some ordinary metric spaces, open set , continuity of reflection.

When we get a set of elements, we want to study their relationship which can be measured by a simple conception (farther or near). If two elements is near that we think they may have more probability to have the same properties. How to measure the ''distance''? As we all know, the Euclidean space $R^n$ with the metric $l^2$. We can expand the metric by four rules. We will give some concrete examples in follow.

Then we want to expand the relationship to a more abstract space (Topology space $\mathcal{T}$) without metric, but some open set.

Finally we will talk something about the reflection T and the continuity which is important for us to deal with different questions in different spaces. (Some questions will be simplified by dealing in another space)