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Linear Form

A linear form (also called as linear functional) is a functional that is linear.

Let $l$ be a linear functional on vector space $X$, then

  1. $l(\alpha x) = |\alpha|l(x), \forall x\in X, \forall \alpha\in\mathbb{K}$
  2. $l(x + y) = l(x) + l(y), \forall x, y\in X$

This is very similar to our usual definition of linear functions but we restrict(generalize) the discussion to vectors.