Consider a pair of [vectors](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors), $\mathbf{x},\mathbf{y}$ belonging to a [vector space](Vector%20spaces.md), $\mathcal{V}.$ The _inner product_, also called a _scalar product,_ is a symmetric [bilinear form](Bilinear%20form.md), $(.,.),$ defined on $\mathcal{V}$ as $\mathcal{V}\times\mathcal{V}\rightarrow\mathbb{F}$ with the following properties that form a definition: 1) $(\mathbf{x}+\mathbf{y}, \mathbf{z}) = (\mathbf{x}, \mathbf{z}) + (\mathbf{y}, \mathbf{z})$ and $(\alpha \mathbf{x}, \mathbf{z}) = \alpha ( \mathbf{x}, \mathbf{z})$ ([bilinearity](Bilinear%20form.md#bilinearity)) ^ad3176 2) $( \mathbf{x}, \mathbf{y}) = ( \mathbf{y}, \mathbf{x})$ (i.e. it is [symmetric](Bilinear%20form.md#Symmetry%20properties%20of%20bilinear%20forms)) ^65abf4 3) $(\mathbf{x},\mathbf{x})>0$ if $x \neq 0$ (positive definiteness) ^c62c47 Additional properties listed below [emerge](Inner%20products.md#Additional%20properties%20of%20inner%20products) from the definition. [property 3](Inner%20products.md#^c62c47) of the definition may also be split into two properties where by _positivity_ $(\mathbf{x},\mathbf{x})\geq 0$ and by _definiteness_ $(\mathbf{x},\mathbf{x})=0 \,\,\,\mathrm{if}\,\mathrm{and}\,\mathrm{only}\,\mathrm{if}\,\,\,\mathbf{x}=0$ %%Interesting to note again here that the existence of an inner product also implies for their being a vector space. Could it be that every inner product has an associated vector space, but not every vector space has an associated inner product? In addition makes sure to check at some point of complex inner products are also bilinear forms.%% # Additional properties of inner products The inner product in a [vector space](Vector%20spaces.md), $\mathcal{V}$ with $\mathbf{x},\mathbf{y},\mathbf{z}\in\mathcal{V}$ fulfills the following additional properties following directly from the [definition](Inner%20products.md) 1) $(0,\mathbf{x})=0$ 2) $(\mathbf{x},\beta \mathbf{y})=\beta^*(\mathbf{x},\mathbf{y})$ where $\beta$ is in $\mathbb{F}$ and $\beta^*$ is the conjugate of $\beta.$ 3) $(\mathbf{x},\alpha\mathbf{y}+\beta\mathbf{z})=\alpha^*(\mathbf{x},\mathbf{y})+\beta^*(\mathbf{x},\mathbf{z})$ where $\alpha$ and $\beta$ are in $\mathbb{F}$ and $\alpha^*$ and $\beta^*$ are their conjugates. 4) For an Operator $T$ on $\mathcal{V}$, $( Tx, x) = 0$ implies $T=0.$ # Inner product spaces In general we refer to any vector space equipped with an [[Inner products]] as an _inner product space._ The simplest inner product space is the [Hausdorff pre-Hilbert space](Hausdorff%20pre-Hilbert%20Space.md), where the only requirement is for an [inner product](Inner%20products.md) to be defined for that [vector space](Vector%20spaces.md). ^7b1847 The existence of an [inner product](Inner%20products.md) also implies the existence of a [vector norm.](Inner%20products.md#vector%20Norms) # Real inner products The [inner product](Inner%20products.md) is equivalent to the _[dot product](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#dot%20product)_ when defined on $\mathbb{R}^n$ for a [real vector space](Real%20vector%20spaces.md) and is expressed as $(\mathbf{x},\mathbf{y})=\mathbf{x}\cdot\mathbf{y} = \sum_j^n x_j y_j.$ ^73cfbe # Complex inner products A complex [[Inner products]], sometimes referred to as a _Hermitian product_, is often expressed with the following notation: $\langle.,.\rangle$. This is again a [bilinear form](Bilinear%20form.md) and it is defined on $\mathbb{C}^n$ for a [complex inner product space,](Complex%20vector%20spaces.md#Complex%20inner%20product%20spaces) $\mathcal{V},$ such that for all vectors $|x\rangle$ and $|y\rangle$ in $\mathcal{V},$ $\langle x,y \rangle = \sum_j x^*_j y_j.$ ^519c62 where $x_j$ and $y_j$ are the elements of vectors $|x\rangle$ and $|y\rangle.$ ^a398e5 For complex inner products [property 2](Inner%20products.md#^65abf4) from the definition of inner products may be rewritten as $\langle x,y \rangle=\langle x,y \rangle^*$ A [complex vector space with an inner product](Complex%20vector%20spaces.md#Complex%20inner%20product%20spaces) is sometimes referred to as a _unitary space_. ^ceeab3 %%Does the inner product imply the existence of a vector space and is a complex inner product also a bilinear form?%% ## vector Norms [[vector norm]] --- # Recommended reading An introduction to the [inner product](Inner%20products.md) is found in practically any linear algebra textbook. In particular see: * [Axler S., Gerhing F.W., Ribet K.A. _Linear Algebra Done Right_, Springer, 2nd edition, 1997](Axler%20S.,%20Gerhing%20F.W.,%20Ribet%20K.A.%20Linear%20Algebra%20Done%20Right,%20Springer,%202nd%20edition,%201997.md) pgs. 97-101. In particular, here, the notion of the inner product is motivated by the notion of vector norms on both real and complex vector spaces. The properties that are considered [additional properties](Inner%20products.md#Additional%20properties%20of%20inner%20products) are derived from properties given here. As the inner product is of key importance in quantum mechanics, it is also introduced in many introductory texts in quantum mechanics as well as in mathematical physics textbooks. * [Griffiths D. J., _Introduction to Quantum Mechanics_, Pearson Prentice Hall, 2nd edition, 2005.](Griffiths%20D.%20J.,%20Introduction%20to%20Quantum%20Mechanics,%20Pearson%20Prentice%20Hall,%202nd%20edition,%202005..md) pg. 94. Here the inner product is defined as a generalization of of the dot product for complex vectors. In this context this means complex vectors in Hilbert Spaces. * [von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md) pgs. 27-28. Here the inner product is defined in terms of [complex vector spaces](Complex%20vector%20spaces.md) namely [Hilbert spaces.](Hilbert%20Space.md) For an in-depth discussion of inner products and the surrounding mathematics see: * [Istrăţescu, V. I., _Inner Product Structures: Theory and Applications_, Springer 1987]([Mathematics%20and%20its%20Applications%2025]%20Vasile%20Ion%20Istrăţescu%20(auth.)%20-%20Inner%20Product%20Structures_%20Theory%20and%20Applications%20(1987,%20Springer%20Netherlands)%20-%20libgen.lc.pdf) pg. 111 gives a definition of inner product spaces around 4 fundamental properties that are the basis for the definition used here. The remainder of this text is dedicated to inner products and inner product spaces. #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra