A Euclidean [space](Spaces.md) is the space within which _[Euclidean geometry](Geometry%20(index).md#Euclidean%20Geometry)_ (i.e. the geometry of straight lines and planes) is done. The term Euclidean Space may be precisely defined in a couple of slightly different ways: * as the [real vector space](Real%20vector%20spaces.md), $\mathcal{E},$ over $\mathbb{R^n}$. This space is also an [inner product space](Inner%20products.md#Inner%20product%20spaces) where the inner product is equivalent to the so called _[dot product](Inner%20product.md#Real%20inner%20product)_. * An [affine space](Affine%20space.md) $\mathbb{E}^n$ - if we were too choose a coordinate system with an _origin_ then this affine space becomes a [vector space](Vector%20spaces.md). ^e6148f # Euclidean vector space %%Here talk about how associated with euclidean vector spaces are sets of points in an affine space - and define the notion of a euclidean vector space%% # Inner product Given two vectors, $\mathbf{u},\,\mathbf{v}\in \mathcal{V}$ where $\mathcal{V}$ is a Euclidian space the [inner product](Inner%20products.md) is the _dot product_ defined as $\langle u,v \rangle = \sum_j u_j v_j =\mathbf{u}^T\mathbf{v}=\mathbf{u}\cdot\mathbf{v}.$ where we took the [matrix transpose](Transpose%20of%20a%20linear%20map.md#transpose%20of%20a%20matrix) of the vector $\mathbf{u}$. # Relation to Hilbert Spaces [Hilbert spaces](Hilbert%20Space.md) are a [generalization of a Euclidean spaces](Hilbert%20Space.md#Generalization%20of%20Euclidean%20Space) thus all Euclidean spaces are also Hilbert spaces. %%Define exactly how the Hilbert space is a generalization of Euclidean spaces%% ## Examples of Euclidean Spaces * The space that models physical space (excluding time) within which known living creatures inhabit is a Euclidean space in $\mathbb{R}^3$. ^06b56c * The modeling of light beams as rays in $\mathbb{R}^2$ under the [paraxial approximation](Paraxial%20approximation.md). * the [$L^2(I)$](L2(I)%20space.md), which consists of square-integrable functions $f(x)$ where $x\in \mathbb{R}.$ --- # Recommended reading %%The definition of Euclidean spaces in von Delft's text on page 47 is confusing. It seems to imply all inner product spaces are Euclidean spaces and I want to be sure it's true. What's the essence of Euclidean spaces? It would seem that its their flatness and that's it but I've never seen a text refer to a complex inner product space as Euclidean%% #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces #MathematicalFoundations/Geometry