Continuous Vector Space

A Continuous Vector Space is a vector space with continuous vector operations (on continuous-valued vectors).

• AKA: Real Vector Space, Rⁿ.
• Context:
  • It can range from being a Normed Real Vector Space to being an Unnormed Vector Space.
  • It can follow a Continuous Vector Space Pattern.
  • …
• Example(s):
  • a Continuous Word Vector Space.
  • …
• Counter-Example(s):
  • a Discrete Vector Space.
  • a Complex Vector Space.
• See: Vector Space Basis, Metric Space.

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References

2013

• http://mathworld.wolfram.com/RealVectorSpace.html
  • QUOTE: A real vector space is a vector space whose field of scalars is the field of reals. A linear transformation between real vector spaces is given by a matrix with real entries (i.e., a real matrix).

2012

• (Golub & Van Loan, 2012) ⇒ Gene H. Golub, and Charles F. Van Loan. (2012). “Matrix Computations (4th Ed.)." Johns Hopkins University Press. ISBN:1421408597
  • QUOTE: Let [math]\displaystyle{ \R^n }[/math] denote the vector space of real n-vectors: :[math]\displaystyle{ x \in \mathbb{R}^n \Leftrightarrow \ \ x = \begin{bmatrix} x_{1} \\ \vdots \\ x_n \end{bmatrix} \ \ x_i \in \R. }[/math] We refer to [math]\displaystyle{ x_i }[/math] as the ith component of [math]\displaystyle{ x }[/math]. Depending upon context, the alternative notations [math]\displaystyle{ [x]_i }[/math] and [math]\displaystyle{ x(i) }[/math] are sometimes used.

2003

• (Gartner, 2003) ⇒ Thomas Gärtner. (2003). “A Survey of Kernels for Structured Data.” In: ACM SIGKDD Explorations Newsletter, 5(1).
  • QUOTE: Usually, to apply kernel methods to 'real-world' data, extensive pre-processing is performed to embed the data into a real vector space and thus in a single table. This survey describes several approaches of defining positive definite kernels on structured instances directly.

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