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**Definition 1** A vector space over the number field or is called an *inner product space* if it is equipped with an *inner product* satisfying for all and scalar ,

- , if and only if ,
- ,
- , and
- .

is an inner product space over with the inner product

An inner product space over is usually called a *Euclidean space.*

The following properties of an inner product can be deduced from the four axioms in Definition 1:

- ,
- ,
- ,
- for all if and only if , and
- .

An important property shared by all inner products is the *Cauchy-Schwarz inequality* and, for an inner product space, one of the most useful inequalities in mathematics.

**Theorem 1**(Cauchy-Schwarz Inequality) Let be an inner product space. Then for all vectors and in over the field or ,

Equality holds if and only if and are linearly dependent.

The proof of this can be done in a number of different ways. The most common proof is to consider the quadratic function in

and derive the inequality from the non-positive discriminant. We will first present this proof.

**Proof:** Let be given. If , the assertion is trivial, so we may assume that . Let and consider

which is a real quadratic polynomial with real coefficients. Because of axiom (1.), we know that for all real , and hence can have no real simple roots. The discriminant of must therefore be non-positive

and hence

Since this inequality must hold for any pair of vectors, it must hold if is replaced by , so we also have the inequality

But , so

If , then the statement of the theorem is trivial; if not, then we may divide equation (2) by the quantity to obtain the desired inequality

Because of axiom (1.), can have a real (double) root only if for some . Thus, equality can occur in the discriminant condition in equation (1) if and only if and are linearly dependent.

We will now present a matrix proof, focusing on the complex vector space, which is perhaps the simplest proof of the Cauchy-Schwarz inequality.

**Proof:** For any vectors we noticed that,

By taking the determinant for the matrix,

the inequality follows at once,

Equality occurs if and only if the matrix has rank 1; that is, and are linearly dependent.

**References **

[1] Zhang, Fuzhen. *Matrix theory: basic results and techniques*. Springer Science & Business Media, 2011.

**Euler Formula **

The Euler formula, sometimes also called the Euler identity, states

where is the imaginary unit. Note that Euler’s polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula.

The special case of the formula with gives the beautiful identity

an equation connecting the fundamental numbers and , the fundamental operations , , and exponentiation, the most important relation , and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.