There are several recommended books for mathematical foundations for UPSC Mathematics Optional. These books provide a comprehensive understanding of the subject and cover all the necessary topics required for the UPSC examination. Here are some of the recommended books:

1. "Mathematical Analysis" by S.C. Malik and Savita Arora:
- This book covers topics such as limits, continuity, differentiation, integration, and sequences and series.
- It provides detailed explanations and numerous examples to help students grasp the concepts effectively.
- The book also includes practice exercises and solved questions to enhance problem-solving skills.

2. "Linear Algebra" by K.C. Prasad, K.B. Datta, and P.N. Chatterjee:
- This book focuses on topics like vector spaces, linear transformations, eigenvalues, eigenvectors, and diagonalization.
- It provides a clear and concise explanation of concepts with numerous solved examples and practice exercises.
- The book also includes previous year question papers to help students understand the exam pattern and types of questions asked in the UPSC examination.

3. "Differential Equations" by M.D. Raisinghania:
- This book covers various types of differential equations, including ordinary differential equations, partial differential equations, and applications of differential equations.
- It provides step-by-step solutions to problems and includes numerous examples to illustrate the concepts.
- The book also includes previous year question papers and practice exercises to enhance problem-solving skills.

4. "Real Analysis" by S.C. Malik and Savita Arora:
- This book covers topics such as sets and functions, limits, continuity, differentiation, integration, and sequences and series.
- It provides a rigorous and comprehensive understanding of real analysis with detailed explanations and examples.
- The book includes practice exercises and solved questions to help students practice and revise the concepts effectively.

In addition to these books, it is also recommended to refer to previous year question papers and solve them to get a better understanding of the exam pattern and types of questions asked in the UPSC examination. Regular practice and revision are essential to excel in the mathematics optional subject for the UPSC examination.