Use the Cauchy-Schwarz inequality to prove the triangle inequality, which states $$ \|x+y\| \leq\|x\|+\|y\| $$ Explain why this is called the triangle inequality.

The Cauchy-Schwarz inequality is a fundamental inequality in linear algebra and vector space theory. It states that for any vectors \(x\) and \(y\) in an inner product space, the absolute value of their inner product is at most the product of their norms. Mathematically, this is expressed as \( |\langle x, y \rangle| \leq \|x\| \|y\| \).
This inequality is very powerful because it provides a bound on the inner product, which is a measure of how much two vectors point in the same direction.

The Cauchy-Schwarz inequality is crucial for proving other important results, such as the triangle inequality. It helps in comparing magnitudes and directions of vectors, which is essential in many areas of mathematics and physics.

A vector norm is a function that assigns a length or size to a vector. The norm of a vector \( x \), denoted by \( \|x\| \), is a measure of its magnitude. For any vector \( x \), its norm is defined as \( \|x\| = \sqrt{\langle x, x \rangle} \), where \( \langle x, x \rangle \) is the inner product of the vector with itself.
Norms are used to determine the distance between vectors and to measure the size of vectors in various applications.

There are different types of norms, but the one discussed here is the Euclidean norm (also known as the 2-norm). It is widely used because it corresponds to our intuitive idea of length in a Euclidean space.

• For instance, in 3D space, the norm of a vector \( x = (x_1, x_2, x_3) \) is \( \|x\| = \sqrt{x_1^2 + x_2^2 + x_3^2} \).
• Norms must satisfy certain properties: non-negativity, definiteness, scalability, and the triangle inequality.

Understanding vector norms is essential for applications in physics, engineering, computer science, and more.

The inner product is a mathematical operation that takes two vectors and returns a scalar. It is denoted by \( \langle x, y \rangle \) and is sometimes called the dot product in Euclidean space.
The inner product of two vectors \( x \) and \( y \) is given by \( \langle x, y \rangle = \sum_{i=1}^n x_i y_i \) in the case of real-valued vectors.

This value measures how much one vector extends in the direction of another and can be thought of as a projection.

• An important property of the inner product is its linearity, which means it satisfies \( \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle \) and \( \langle \alpha x, y \rangle = \alpha \langle x, y \rangle \) for any scalar \( \alpha \).
• Another property is symmetry: \( \langle x, y \rangle = \langle y, x \rangle \).

Understanding the inner product is crucial for many areas of mathematics and its applications because it sets the foundation for defining orthogonality, angle between vectors, and projection concepts.