In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.
An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An incomplete space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm, induced by the inner product, becomes a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.
Read more about Inner Product Space: Definition, Examples, Norms On Inner Product Spaces, Orthonormal Sequences, Operators On Inner Product Spaces, Generalizations, Related Products
Famous quotes containing the words product and/or space:
“A product of the untalented, sold by the unprincipled to the utterly bewildered.”
—Al Capp (1909–1979)
“And Space with gaunt grey eyes and her brother Time
Wheeling and whispering come,”
—James Elroy Flecker (1884–1919)