Inner product

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In mathematics, an inner product is an abstract notion on general vector spaces that is a generalization of the concept of the dot product in the Euclidean spaces. Among other things, the inner product on a vector space makes it possible to define the geometric operation of projection onto a closed subspace (in the metric topology induced by the inner product), just like how the dot product makes it possible to define, in the Euclidean spaces, the projection of a vector onto the subspace spanned by a set of other vectors. The projection operation is a powerful geometric tool that makes the inner product a desirable convenience, especially for the purposes of optimization and approximation.

## Formal definition of inner product[edit]

Let X be a vector space over a sub-field F of the complex numbers. An inner product  ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on X is a sesquilinear[1] map from  X × X {\displaystyle X\times X} to  C {\displaystyle \mathbb {C} } with the following properties:

1. ⟨ x , y ⟩ = ⟨ y , x ⟩ ¯ ∀ x , y ∈ X {\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}\,\,\forall x,y\in X}
2. ⟨ x , y ⟩ = 0 ∀ y ∈ X ⇒ x = 0 {\displaystyle \langle x,y\rangle =0\,\,\forall y\in X\Rightarrow x=0}
3. ⟨ α x 1 + β x 2 , y ⟩ = α ⟨ x 1 , y ⟩ + β ⟨ x 2 , y ⟩ {\displaystyle \langle \alpha x_{1}+\beta x_{2},y\rangle =\alpha \langle x_{1},y\rangle +\beta \langle x_{2},y\rangle } ∀ α , β ∈ F {\displaystyle \forall \alpha ,\beta \in F} and  ∀ x 1 , x 2 , y ∈ X {\displaystyle \forall x_{1},x_{2},y\in X} (linearity in the first slot)
4. ⟨ x , α y 1 + β y 2 ⟩ = α ¯ ⟨ x , y 1 ⟩ + β ¯ ⟨ x , y 2 ⟩ {\displaystyle \langle x,\alpha y_{1}+\beta y_{2}\rangle ={\bar {\alpha }}\langle x,y_{1}\rangle +{\bar {\beta }}\langle x,y_{2}\rangle } ∀ α , β ∈ F {\displaystyle \forall \alpha ,\beta \in F} and  ∀ x , y 1 , y 2 ∈ X {\displaystyle \forall x,y_{1},y_{2}\in X} (anti-linearity in the second slot)
5. ⟨ x , x ⟩ ≥ 0 ∀ x ∈ X {\displaystyle \langle x,x\rangle \geq 0\,\,\forall x\in X} (in particular it means that  ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is always real)
6. ⟨ x , x ⟩ = 0 ⇒ x = 0 {\displaystyle \langle x,x\rangle =0\Rightarrow x=0}

Properties 1 and 2 imply that  ⟨ x , y ⟩ = 0 ∀ x ∈ X ⇒ y = 0 {\displaystyle \langle x,y\rangle =0\,\forall x\in X\Rightarrow y=0} .

Note that some authors, especially those working in quantum mechanics, may define an inner product to be anti-linear in the first slot and linear in the second slot, this is just a matter of preference. Moreover, if F is a subfield of the real numbers  R {\displaystyle \mathbb {R} } then the inner product becomes a bilinear map from  X × X {\displaystyle X\times X} to  R {\displaystyle \mathbb {R} } , that is, it becomes linear in both slots. In this case the inner product is said to be a real inner product (otherwise in general it is a complex inner product).

## Norm and topology induced by an inner product[edit]

The inner product induces a norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on X defined by  ‖ x ‖ = ⟨ x , x ⟩ 1 / 2 {\displaystyle \|x\|=\langle x,x\rangle ^{1/2}} . Therefore it also induces a metric topology on X via the metric  d ( x , y ) = ‖ x − y ‖ {\displaystyle d(x,y)=\|x-y\|} .

## Reference[edit]

1. ↑ T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York (1982), ISBN 0-387-90666-5 p. 49