Normed Vector Space

A Normed Vector Space is a vector space equipped with a norm function that assigns a non-negative real number to each vector, representing its length or magnitude.

• AKA: Normed Linear Space, Normed Space, NVS.
• Context:
  • It can measure Vector Length through its normed vector space norm function.
  • It can induce Distance Metric Space through its normed vector space distance function.
  • It can support Linear Operations while preserving topological properties.
  • It can enable Convergence Analysis through its normed vector space topology.
  • It can facilitate Approximation Theory through its normed vector space structure.
  • It can normalize Vectors to create Unit Vectors and Normalized Vectors.
  • ...
  • It can often serve as Mathematical Frameworks for functional analysis.
  • It can often model Physical Quantities in applied mathematics.
  • It can often provide Geometric Intuition for abstract mathematical concepts.
  • It can often support Optimization Methods through its normed vector space gradient structure.
  • It can often enable Regularized Learning Tasks through norm-based regularization.
  • It can often support Regularized Linear Regression Tasks through normed vector space penalty terms.
  • ...
  • It can range from being a Finite-Dimensional Normed Vector Space to being an Infinite-Dimensional Normed Vector Space, depending on its normed vector space dimensional characteristic.
  • It can range from being a Complete Normed Vector Space to being an Incomplete Normed Vector Space, depending on its normed vector space Cauchy sequence behavior.
  • It can range from being a Separable Normed Vector Space to being a Non-Separable Normed Vector Space, depending on its normed vector space density property.
  • It can range from being a Normed Real Vector Space to being a Normed Complex Vector Space, depending on its normed vector space scalar field.
  • ...
  • It can satisfy Norm Axioms including positive definiteness, absolute homogeneity, and triangle inequality.
  • It can be embedded in Complete Normed Vector Spaces through completion processes.
  • It can generate Topological Vector Space Structures through its normed vector space induced topology.
  • It can be induced by Inner Product Spaces through the inner product norm.
  • It can define Vector Length Functions for vector magnitude computation.
  • ...
• Examples:
  • Complete Normed Vector Spaces, such as:
    • Banach Spaces, such as:
      • ℓᵖ Space for normed vector space sequence analysis.
      • Lᵖ Space for normed vector space function analysis.
    • Hilbert Spaces (which are Inner Product Spaces), such as:
      • L² Space for normed vector space square-integrable functions.
      • Euclidean Space for normed vector space finite-dimensional geometry.
  • Incomplete Normed Vector Spaces, such as:
    • Polynomial Spaces, such as:
      • [[C[0,1] Polynomial Subspace]] for normed vector space polynomial approximation.
      • Finite Fourier Series Space for normed vector space trigonometric approximation.
    • Pre-Completion Spaces, such as:
      • Rational Function Space for normed vector space rational approximation.
      • Step Function Space for normed vector space discrete approximation.
  • Continuous Vector Spaces (as Normed Real Vector Spaces), such as:
    • [[C([a,b]) Space]] for normed vector space continuous functions.
    • C^k Space for normed vector space differentiable functions.
  • ...
• Counter-Examples:
  • Pseudo-Normed Spaces, which violate normed vector space positive definiteness.
  • Semi-Normed Spaces, which allow non-zero vectors with zero norm.
  • Metric Spaces without vector space structure, which lack normed vector space linear operations.
  • Topological Vector Spaces without norm function, which lack normed vector space metric structure.
• See: Vector Space, Norm Function, Complete Normed Vector Space, Distance Metric Space, Inner Product Space, Hilbert Space, Unit Vector, Normalized Vector, Vector Length Function, Mathematical Space, Functional Analysis, Real Vector Space, Triangle Inequality, Linear Algebra, Topological Vector Space, Banach Space Theory, Regularized Learning Task.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Normed_vector_space Retrieved:2015-2-7.
  • In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rⁿ. The following properties of "vector length" are crucial.

    1. The zero vector, 0, has zero length; every other vector has a positive length. :[math]\displaystyle{ \|x\|\gt 0 }[/math] if [math]\displaystyle{ x\ne0 }[/math]

    2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
     :[math]\displaystyle{ \|\alpha x\|=|\alpha| \|x\| }[/math] for any scalar [math]\displaystyle{ \alpha. }[/math]

    3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. :[math]\displaystyle{ \|x+y\| \le \|x\|+\|y\| }[/math] for any vectors x and y. (triangle inequality)

    The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is then called a normed vector space.

    Normed vector spaces are central to the study of linear algebra and functional analysis.

1997

• (Luenberger, 1997) ⇒ David G. Luenberger. (1997). “Optimization by Vector Space Methods." Wiley Professional. ISBN:047118117X

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