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 ${\displaystyle ⟨\cdot ,\cdot ⟩}$ on X is a sesquilinear^[1] map from ${\displaystyle X\times X}$ to ${\displaystyle \mathbb{ℂ}}$ with the following properties:

1. ${\displaystyle ⟨x,y⟩=\overline{⟨y,x⟩}\mathrm{\forall }x,y\in X}$
2. ${\displaystyle ⟨x,y⟩=0\mathrm{\forall }y\in X\Rightarrow x=0}$
3. ${\displaystyle ⟨\alpha x_1+\beta x_2,y⟩=\alpha ⟨x_1,y⟩+\beta ⟨x_2,y⟩}$ ${\displaystyle \mathrm{\forall }\alpha ,\beta \in F}$ and ${\displaystyle \mathrm{\forall }{x}_1,x_2,y\in X}$ (linearity in the first slot)
4. ${\displaystyle ⟨x,\alpha y_1+\beta y_2⟩=\overline{\alpha }⟨x,y_1⟩+\overline{\beta }⟨x,y_2⟩}$ ${\displaystyle \mathrm{\forall }\alpha ,\beta \in F}$ and ${\displaystyle \mathrm{\forall }x,y_1,y_2\in X}$ (anti-linearity in the second slot)
5. ${\displaystyle ⟨x,x⟩\ge 0\mathrm{\forall }x\in X}$ (in particular it means that ${\displaystyle ⟨x,x⟩}$ is always real)
6. ${\displaystyle ⟨x,x⟩=0\Rightarrow x=0}$

Properties 1 and 2 imply that ${\displaystyle ⟨x,y⟩=0\mathrm{\forall }x\in X\Rightarrow y=0}$.

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 ${\displaystyle \mathbb{ℝ}}$ then the inner product becomes a bilinear map from ${\displaystyle X\times X}$ to ${\displaystyle \mathbb{ℝ}}$, 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 ${\displaystyle \Vert \cdot \Vert }$ on X defined by ${\displaystyle \Vert x\Vert =⟨x,x{⟩}^{1/2}}$. Therefore it also induces a metric topology on X via the metric ${\displaystyle d(x,y)=\Vert x-y\Vert }$.

Reference[edit]