An inner product space is a vector space with an ** inner product**.

An inner product is an operation that assigns a scalar to a pair of vectors or a scalar to a pair of functions . The way to assign the scalar may be through the matrix multiplication of the pair of vectors, e.g.

or it may be through an integral of the pair of functions.

You may notice that eq3 resembles a ** dot product**. The dot product pertains to vectors in , where , which can be extended to -dimensions, where , and to include complex and real functions, . Therefore, an inner product is a generalisation of the dot product.

An inner product space has the following properties:

- Conjugate symmetry:
- Additivity:
- Positive semi-definiteness: , with if

###### Question

i) Why is the inner product space positive semi-definite? ii) Show that orthogonal vectors are linearly independent.

###### Answer

i) A general vector space of can be positive or negative. The inner product space is defined such that , with if , which is useful in quantum mechanics.

ii) Let the set of vectors in eq1 be orthogonal vectors. The dot product of eq1 with ** **gives . Since the magnitudes of orthogonal vectors are non-zero, . Hence, orthogonal vectors are linearly independent.

Two functions (or two vectors) are ** orthogonal** if . Elements of a set of basis functions are orthonormal if where

In other words, two functions (or two vectors) are ** orthonormal** if they are orthogonal and normalised.

Finally, the norm (or length) of a vector is denoted by and is defined as . With this association of inner product and the length of a vector, we can establish the relationship between inner product and the Euclidean distance between 2 vectors and . Using the space as an example, where and ,** **we have