The uniform boundedness theorem in b-Banach space

B-Banach space is an extension of Banach space, which provides a suitable framework for studying many analytical problems. The uniform boundedness theorem is is the basic theorem in functional analysis and has many important applications in many field, such as matrix analysis, operator theory, and numerical analysis. In this note, we revisit the concept of b-Banach space, and then establish the uniform boundedness theorem for linear operators. The result may be useful to establish linear operator theory in b-Banach space