In the praise of norm


If you have spent some time with undergraduate mathematics, you would have probably heard the word “norm”. This term is encountered in various branches of mathematics, like (as per Wikipedia):

But, it seems to occur only in abstract algebra. Although the definition of this term is always algebraic, it has a topological interpretation when we are working with vector spaces.  It secretly connects a vector space to a topological space where we can study differentiation (metric space), by satisfying the conditions of a metric.  This point of view along with an inner product structure, is explored when we study functional analysis.

Some facts to remember:

  1. Every vector space has a norm. [Proof]
  2. Every vector space has an inner product (assuming “Axiom of Choice”). [Proof]
  3. An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space.  [Proof]
  4. All norms are equivalent in finite dimensional vector spaces. [Proof]
  5. Every normed vector space is a metric space (and NOT vice versa). [Proof]
  6. In general, a vector space is NOT same a metric space. [Proof]

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