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'Euclidean vector space'
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| Title of object: |
Euclidean vector space |
| Canonical Name: |
EuclideanVectorSpace2 |
| Type: |
Definition |
| Created on: |
2006-01-22 12:27:54 |
| Modified on: |
2006-01-22 12:54:12 |
| Classification: |
msc:15A63 |
Revision comment (for changes between this and next version):
| change "analoguos" to "analogous" |
Preamble:
\usepackage{amsmath}
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\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
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\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{theorem}[proposition]{Theorem} |
Content:
\section{Definition}
A Euclidean vector space is a synonym for ''finite-dimensional, real, positive-definite, inner product space''. The canonical example is $\reals^n$ equipped with the usual dot product. Indeed, every Euclidean vector space $V$ is isomorphic to $\reals^n$, up to a choice of orthonormal basis of $V$. As well, every Euclidean vector space $V$ carries a natural metric space structure given by
$$ d(u,v) = \sqrt{\left< u-v, u-v \right>},\quad u,v\in V.$$
\section{Remarks.}
\begin{itemize}
\item An analoguos object with complex numbers as the base field is called a unitary space.
\item Dropping the assumption of finite-dimensionality we arrive at the class of real Hilbert spaces.
\item If we drop the inner product and the vector space structure, but retain the metric space structure, we arrive at the notion of a Euclidean space.
\end{itemize} |
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