Every time I here it mentioned it is praised in the highest posible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only consequences of it that I have read is that it proves that there are lots of functionals and that separating hyperplanes exist. Are those 2 consequences really that spectacular, or are there other ones that I don't know of?

Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only consequences of it that I have read is that it proves that there are lots of functionals and that separating hyperplanes exist. Are those 2 consequences really that spectacular, or are there other ones that I don't know of?