Mathematical Methods for Theoretical Physics

In this section, we will review the mathematical methods needed for theoretical physics. This includes topics such as linear algebra, calculus, differential equations, complex analysis, group theory, differential geometry, topology, and more. We will also cover some more advanced topics such as functional analysis, distribution theory, and category theory Interestingly, much of the mathematics here revolve around vector spaces. Therefore, we will start with a review of vector spaces and linear algebra.

For this section we will follow Mathematical Physics: A Modern Introduction to Its Foundations by Sadri Hassani, which is an excellent book that covers all the necessary mathematics needed for theoretical physics.

As did Hassani, I have included a table for the various symbols used in this section. Some of them might differ from the book.

Symbol	Meaning
General Symbols
	For all
	There exists
iff	If and only if
	is defined to be
	Implies
	The set of natural numbers
	The set of integers
	The set of rational numbers
	The set of real numbers
	The set of complex numbers; ${a + bi
	The set of quaternions; ${a + bi + cj + dk
Set-theoretic Symbols
	Element of
	Not an element of
	Subset of
	Subset of or equal to
	The empty set
	Union
	Intersection
	Complement of set
	Cartesian product of sets and ; ${(a,b)
	Power set of ; the set of all subsets of
Maps and Relations
	A relation
	An equivalence relation
	The equivalence class of
	A map from set to set
	is the image of under the map
	The identity map on set
	The composition of maps and
	The kernel of the map ; ${a \in A
	The image of the map ;
Linear Algebra and Abstract Algebra
	A vector space
	The dual space of the vector space
	A vector in a vector space (Dirac notation)
	A covector/covariant vector/dual vector/linear functional/1-form/bra vector in the dual space (Dirac notation)
	The inner product of vectors and (Dirac notation)
	The norm of vector
	The -dimensional real vector space
	The -dimensional complex vector space
	The set of all matrices with real entries
	The set of all matrices with complex entries
	The set of absolutely convergent real sequences
	The set of absolutely convergent complex sequences
	The set of all complex polynomials in the variable
	The set of all real polynomials in the variable
	The set of all complex polynomials in the variable of degree at most
	The set of all real polynomials in the variable of degree at most
	The set of all continuous functions on the (real) interval
	The set of all -times continuously differentiable functions on the (real) interval
	The set of all infinitely differentiable (smooth) functions on the (real) interval
	The span of the set ; the set of all finite linear combinations of elements of
	Direct sum of vector spaces
	Direct sum of multiple vector spaces
	Tensor product of vector spaces
	Tensor product of multiple vector spaces
	The dual pairing of vector and covector
	An algebra over a field
	The algebra of all linear maps from the vector space to itself
	The algebra of all linear maps from the vector space to itself
	Direct sum of algebras
	Direct sum of multiple algebras
	Tensor product of algebras
	Tensor product of multiple algebras
	The algebra of all square matrices with entries from the field
	The Clifford algebra of the inner product space
	The Clifford product

Mathematical Methods for Theoretical Physics

List of Theorems, Definitions, etc.

1.1 Sets

2.1 Vector Spaces

2.2 Inner Product Spaces

2.3 Linear Maps

2.4 Complex Structures

List of Theorems, Definitions, etc.​

1.1 Sets​

2.1 Vector Spaces​

2.2 Inner Product Spaces​

2.3 Linear Maps​

2.4 Complex Structures​