Inner Product Spaces

Home > Mathematics > Linear algebra > Inner Product Spaces

An inner product space is a vector space with an additional inner product operation. This operation allows for the definition of distance, angles, and orthogonality in the vector space.

What is an inner product space?

"In mathematics, an inner product space is a real vector space or a complex vector space with an operation called an inner product."

How is the inner product of two vectors defined?

"The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as ."

What is the significance of inner products in geometric notions?

"Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors."

How do inner product spaces generalize Euclidean vector spaces?

"Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates."

What is the relation between inner product spaces and functional analysis?

"Inner product spaces of infinite dimension are widely used in functional analysis."

What are inner product spaces over the field of complex numbers referred to as?

"Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces."

Who introduced the concept of a vector space with an inner product?

"The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898."

What does an inner product naturally induce in an inner product space?

"An inner product naturally induces an associated norm, (denoted |x| and |y| in the picture); so, every inner product space is a normed vector space."

What is the additional condition for an inner product space to be a Hilbert space?

"If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space."

How can an inner product space be extended to a Hilbert space if it is not already one?

"If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space H̄."

What is the relationship between H and H̄ in the extension process to a Hilbert space?

"H is a linear subspace of H̄, the inner product of H is the restriction of that of H̄, and H is dense in H̄ for the topology defined by the norm."

What concepts can be defined using inner products in an inner product space?

"Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors."

How do inner product spaces relate to normed vector spaces?

"Every inner product space is a normed vector space."

What is the importance of completeness in an inner product space?

"If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space."

What notions of vectors can be represented using inner products?

"Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors."

Do inner product spaces only exist in finite dimensions?

"Inner product spaces of infinite dimension are widely used in functional analysis."

How are inner product spaces related to Euclidean vector spaces?

"Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates."

What is the term used to refer to inner product spaces over the field of complex numbers?

"Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces."

What additional structure do inner product spaces inherit from their inner product operation?

"An inner product naturally induces an associated norm, so every inner product space is a normed vector space."

What is the role of inner product spaces in the study of functional analysis?

"Inner product spaces of infinite dimension are widely used in functional analysis."