Euclidean Space

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A Euclidean space enables the investigation of Topological properties such as Compactness . An Inner Product Space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within Functional Analysis . Euclidean space plays a part in the definition of a Manifold which embraces the concepts of both Euclidean and Non-Euclidean Geometry . One mathematical motivation for defining a distance function is the ability to define an Open Ball around points in the space. This fundamental concept justifies a Differential Calculus between a Euclidean space and other manifolds. Differential Geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of non-Euclidean manifolds. REAL COORDINATE SPACE Let R denote the Field of Real Number s. For any non-negative Integer ''n'', the space of all ''n''- Tuple s of real numbers forms an ''n''-dimensional Vector Space over R sometimes called '''real coordinate space''' and denoted R''n''. An element of R''n'' is written '''x''' = (''x''1, ''x''2, …, ''x''''n'') where each ''x''''i'' is a real number. The vector space operations on R''n'' are defined by :$\backslash mathbf\{x\}\; +\; \backslash mathbf\{y\}\; =\; (x\_1\; +\; y\_1,\; x\_2\; +\; y\_2,\; \backslash ldots,\; x\_n\; +\; y\_n)$ :$a\backslash ,\backslash mathbf\{x\}\; =\; (a\; x\_1,\; a\; x\_2,\; \backslash ldots,\; a\; x\_n)$ Real coordinate space R''n'' comes with a Standard Basis : :$\backslash mathbf\{e\}\_1\; =\; (1,\; 0,\; \backslash ldots,\; 0)$ :$\backslash mathbf\{e\}\_2\; =\; (0,\; 1,\; \backslash ldots,\; 0)$ : :$\backslash mathbf\{e\}\_n\; =\; (0,\; 0,\; \backslash ldots,\; 1)$ An arbitrary vector in R''n'' can then be written in the form :$\backslash mathbf\{x\}\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; \backslash mathbf\{e\}\_i$ Real coordinate space is the prototypical example of a real ''n''-dimensional vector space. In fact, every real ''n''-dimensional vector space ''V'' is Isomorphic to R''n''. This isomorphism is not Canonical however. A choice of isomorphism is equivalent to a choice of Basis for ''V'' (by looking at the image of the standard basis for R''n'' in ''V''). The reason for working with arbitrary vector spaces instead of R''n'' is that it is often preferable to work in a ''coordinate-free'' manner (i.e. without choosing a preferred basis). EUCLIDEAN STRUCTURE Euclidean space is more than just real coordinate space. In order to do Euclidean Geometry one needs to be able to talk about the Distance between points and the Angle s between lines or vectors. The natural way in which to do this is to introduce what is called an Inner Product or ''dot product'' on R''n''. This product is defined by :$\backslash mathbf\{x\}\backslash cdot\backslash mathbf\{y\}\; =\; \backslash sum\_\{i=1\}^n\; x\_iy\_i\; =\; x\_1y\_1+x\_2y\_2+\backslash cdots+x\_ny\_n.$ The dot product of any two vectors x and '''y''' gives a real number. This product allows us to define the "length" of a vector ''x'' in the following way

  :<math>d(\mathbf{x}, \mathbf{y}) \\mathbf{x} - \mathbf{y}\ = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}</math>