From computational mathematics to abstract mathematics
For application to a US doctoral program in pure math, I think you would be well situated, other things being equal. This would be especially the case if your chosen field is Analysis or something related. However, the path might take a bit longer than if you had studied a broader spectrum of math. 

The program you would be entering would have two essential requirements (perhaps a few minor ones). The first is to pass a set of comprehensive examinations. These are intended to assure that you have breadth of knowledge. The second is to write an acceptable dissertation. This is to show you have the ability and insight to delve deeply into a narrow aspect of math. There might be a few other options at a few places, but this is the standard. 

One comprehensive exam will probably be in abstract algebra or some sub field. Another might be in Topology. But there will likely be at least three such exams, and one or more of them might be orals. In fact, there might be two levels at some places: written exams and orals. 

But a US doctoral program also normally comes with a fair amount of advanced coursework. This is intended primarily to get you through the qualifying exams. Many US students starting a doctoral program won't have a masters, so the coursework is intended to give them the necessary breadth. 

You may be ready for the Analysis qualifying exams, but would probably need the coursework for any others. It would take some time. If you are ready for that, it might be worth applying. 

Also note that most math doctoral students in US work as TAs. This provides funding for them and assistance in teaching undergraduates. This adds to the time required to earn the degree, of course. 

Note also, that in some cases the pure and applied math departments are separate. In others, they are joined in a single department. It might be a consideration in your case.