Transition from introduction to analysis to more advanced analysis I am currently studying intro to analysis and learning somethings about
basic topology in metric space and almost finished the course . I am
thinking of taking some more advanced
analysis. Would it be demanding to take some course like 
 functional analysis or real analysis only with knowledge of intro analysis course
?Do the courses need more mathmatical knowledge to handle?

Once you have a basic intro to real analysis, you can go several ways.
One would be to cover analysis in $\mathbb{R}^N$ and heading towards analysis on manifolds using books like Rudin's "Principles of Mathematical Analysis" (chapters 9 and 10), or Munkres' "Analysis on Manifolds".
Another direction you can take, if you have covered the Riemann integral in basic analysis, is to study measure and integration theory. You could use the first of Fremlin's free books on measures and integration. I don't have a very good idea of introductory books for this.
If you also know some linear algebra, you could also go on into functional analysis. There are some good books meant for students with a modest background in real analysis and linear algebra. One very popular book is "Introductory Functional Analysis" by Kreyszig. This book neither assumes nor uses Lebesgue integration. If you got as far as sequences of functions in basic analysis, you could also use "Beginning Functional Analysis" by Karen Saxe (this one first gives a self-contained intro to Lebesgue integration).
