inner product

an inner product on a vector space $\text{[math]}$ is an operation $\text{[math]}$ on pairs of vectors in $\text{[math]}$ that satisfies the same conditions that the dot product in euclidean space does: namely, bilinearity, symmetry, and positive definiteness. a vector space equipped with an inner product is an inner product space.
[cite:@abstract_gallian_2021 chapter 1 inner product spaces]

if $\text{[math]}$ , we define the inner product $\text{[math]}$ to be the quantity
$\text{[math]}$ [cite:@tao_analysis_2 definition 5.2.1]

an inner product on $\text{[math]}$ is a function that takes each ordered pair $\text{[math]}$ of elements of $\text{[math]}$ to a number $\text{[math]}$ and has the following properties.

positivity: $\text{[math]}$ for all $\text{[math]}$ .
definiteness: $\text{[math]}$ if and only if $\text{[math]}$ .
additivity in first slot: $\text{[math]}$ for all $\text{[math]}$ .
homogeneity in first slot: $\text{[math]}$ for all $\text{[math]}$ and all $\text{[math]}$ .
conjugate symmetry: $\text{[math]}$ for all $\text{[math]}$ .

[cite:;taken2 from @algebra_axler_2024 definition 6.2]

let $\text{[math]}$ be an inner product on a vector space $\text{[math]}$ . for $\text{[math]}$ and $\text{[math]}$ arbitrary scalars:

$\text{[math]}$ .
$\text{[math]}$ is real and non-negative.
$\text{[math]}$ iff $\text{[math]}$ .
$\text{[math]}$ . this implies that an inner product is a sesquilinear form.
$\text{[math]}$ .

a matrix $\text{[math]}$ defines an inner product $\text{[math]}$ on $\text{[math]}$ iff $\text{[math]}$ is symmetric and broken link: blk:1749579240.7732065. this is equivalent to all eigenvalues being positive.

$\text{[math]}$ . $\text{[math]}$ . (dot product)
$\text{[math]}$ . $\text{[math]}$ . this is a hermitian inner product, not bilinear but sesquilinear. for real vector spaces, $\text{[math]}$ .
$\text{[math]}$ (space of $\text{[math]}$ real matrices). $\text{[math]}$ .
$\text{[math]}$ . $\text{[math]}$ . (Hermitian inner product). $\text{[math]}$ .
$\text{[math]}$ (space of continuous real-valued functions on $\text{[math]}$ ). $\text{[math]}$ . this is symmetric and positive-definite.
$\text{[math]}$ (polynomials). $\text{[math]}$ for some weight function $\text{[math]}$ .

determine if $\text{[math]}$ is an inner product:

$\text{[math]}$ on $\text{[math]}$ .
$\text{[math]}$ on $\text{[math]}$ .
$\text{[math]}$ on $\text{[math]}$ .
$\text{[math]}$ . on (i) $\text{[math]}$ . (ii) $\text{[math]}$ . (iii) $\text{[math]}$ =real integrable functions.
$\text{[math]}$ . on $\text{[math]}$ .

for an inner product space $\text{[math]}$ , for all $\text{[math]}$ : $\text{[math]}$ . equality holds if and only if $\text{[math]}$ and $\text{[math]}$ are linearly dependent (if one vector is a multiple of the other by a scalar).
[cite:;found in @calc_hubbard_2015 chapter 1.4.5]