Inner product space

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In mathematics, an inner product space is a type of vector space. As well as all the normal axioms of a vector space, an inner product space has an added binary operation called the inner product.

The inner product generalises the dot product used in Euclidean spaces, and "dot product" is often used informally to talk about the inner product of a given space.

Definition

An inner product space is defined with two pieces, the vector space ${\displaystyle V}$ and the inner product function. Also important is ${\displaystyle F}$, the field that used to define ${\displaystyle V}$. The inner product takes two vectors from ${\displaystyle V}$ and sends them to ${\displaystyle F}$. The domain of the inner product is commonly written with the Cartesian product:

${\displaystyle \left\langle {\vec {x}},{\vec {y}}\right\rangle :V\times V\to F}$

In order to be an inner product space, the function needs three properties:

• It must be conjugate symmetric, meaning that the inner product of the complex conjugates must be equal to the conjugate of the reversed product:

${\displaystyle \left\langle {\vec {\overline {x}}},{\vec {\overline {y}}}\right\rangle ={\overline {\left\langle {\vec {y}},{\vec {x}}\right\rangle }}}$

• It must be linear in one of its arguments. Both orderings are used: linearity in the first argument is more common in abstract algebra, while linearity in the second argument is more common in physics.

Linear first argument: ${\displaystyle \left\langle a{\vec {x}}+b{\vec {y}},{\vec {z}}\right\rangle =a\left\langle {\vec {x}},{\vec {z}}\right\rangle +b\left\langle {\vec {y}},{\vec {z}}\right\rangle }$

Linear second argument: ${\displaystyle \left\langle {\vec {z}},a{\vec {x}}+b{\vec {y}}\right\rangle =a\left\langle {\vec {z}},{\vec {x}}\right\rangle +b\left\langle {\vec {z}},{\vec {y}}\right\rangle }$

• It must be positive-definite, meaning that the inner product of a vector with itself is never a negative number, and is only equal to zero for the zero vector:

${\displaystyle \left\langle {\vec {x}},{\vec {x}}\right\rangle \geq 0}$

${\displaystyle \left\langle {\vec {x}},{\vec {x}}\right\rangle =0\iff {\vec {x}}={\vec {0}}}$

Matrix definition

Inner product spaces can also be defined using a positive-definite Hermitian matrix and matrix multiplication:

${\displaystyle \left\langle {\vec {x}},{\vec {y}}\right\rangle ={\vec {x}}^{T}M{\vec {y}}}$

From a known inner product, the matrix can be found by taking inner products of a basis of the space. For example, a three-dimensional inner product space with basis vectors ${\displaystyle {\vec {u_{1}}},{\vec {u_{2}}},{\vec {u_{3}}}}$ has the matrix representation $${\displaystyle \left\langle {\vec {x}},{\vec {y}}\right\rangle ={\vec {x}}^{T}\left[{\begin{array}{ccc}\left\langle {\vec {u_{1}}},{\vec {u_{1}}}\right\rangle &\left\langle {\vec {u_{1}}},{\vec {u_{2}}}\right\rangle &\left\langle {\vec {u_{1}}},{\vec {u_{3}}}\right\rangle \\\left\langle {\vec {u_{2}}},{\vec {u_{1}}}\right\rangle &\left\langle {\vec {u_{2}}},{\vec {u_{2}}}\right\rangle &\left\langle {\vec {u_{2}}},{\vec {u_{3}}}\right\rangle \\\left\langle {\vec {u_{3}}},{\vec {u_{1}}}\right\rangle &\left\langle {\vec {u_{3}}},{\vec {u_{2}}}\right\rangle &\left\langle {\vec {u_{3}}},{\vec {u_{3}}}\right\rangle \end{array}}\right]{\vec {y}}}$$ When the basis vectors are orthonormal (perpendicular and ${\displaystyle \left\langle {\vec {u}},{\vec {u}}\right\rangle =1}$), this gives the dot product.