An inner product is a generalization of the dot product. At this point you may be tempted to guess that an inner product is defined by abstracting the properties of the dot product discussed in the last paragraph.

An inner product in the vector space of continuous functions in [0; 1], denoted as V = C([0; 1]), is de ned as follows. Given two arbitrary vectors f(x) and g(x), introduce the inner product

An inner product, also known as dot product or scalar product is an operation on vectors of a vector space which from any two vectors x and y produces a number which we denote by (x, y).

We discuss inner products on nite dimensional real and complex vector spaces. Although we are mainly interested in complex vector spaces, we begin with the more familiar case of the usual inner product.

A vector space Z with an inner product defined is called an inner product space. Because any inner product “acts just like” the inner product from ‘ 8 , many of the theorems we proved about inner …

After this lesson you should be able to: 1Define inner products, 2Understand the relation of inner products to norms, 3Recognize inner products beyond matrices and vectors, 4Construct and utilize …

inner product is a Hilbert space. An infinite-dimensional complex vector space with an inner product is also a Hilbert space if a completeness property holds. This prop-erty is a technical property which is …