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I A Question about Notation and Continuous Linear Functionals

  1. Jul 20, 2016 #1
    I have reading through various sources on linear functionals, but all seem somewhat inconsistent with regard to denoting the set of all linear functionals and the set

    Also, what is the standard definition of a continuous linear functional? I really couldn't find much besides
    this
    Let ##f : V \rightarrow \mathbb{K}## be a linear functional. It is said to be continuous if and only if for every ##x \in V##, there exists a positive number ##M## such that

    ##\frac{|f(x)|}{||x||} \le M##

    where ##||x||## is the norm of ##x##. The least such number ##M## is called the norm of ##F##, written as ##||f||##,

    which comes from http://www.solitaryroad.com/c855.html
     
  2. jcsd
  3. Jul 20, 2016 #2

    RUber

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    That seems right for the definition of a continuous linear functional. Clearly this is dependent on your choice of norm for your function space. Also, it implies that if ##\|x\| = 0, ## then ##f(x) = 0##.

    Was there supposed to be something here?
     
  4. Jul 20, 2016 #3
    Sorry. I thought I had finished the sentence! What I was asking was, what is the standard notation for the set of all linear functionals, and in particular the set of all continuous linear functionals?
     
  5. Jul 20, 2016 #4
    Theorem. A linear function ##f:V\to\mathbb{R}## is continuous iff there is a constant ##C## such that ##|f(x)|\le C\|x\|,\quad \forall x\in V##.
    Exercise: prove this theorem.
     
  6. Jul 20, 2016 #5
    One of the standard notations: ##V'## -- the space of continuous linear functions of the normed space ##V## and ##V^*## -- the space of all linear functions of ##V##
     
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