Results 1 to 3 of 3

Math Help - Prove that y is an element of the dual space of P.

  1. #1
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1

    Prove that y is an element of the dual space of P.

    Hi,

    problem:

    If (\alpha_0,\alpha_1,\alpha_2) is an arbitrary sequence of complex numbers, and if x is an element of \mathbb{P}, x(t)=\sum^n_{i=0}\xi_it^i,
    write y(x)=\sum^n_{i=0}\xi_i\alpha_i. Prove that y is an element of \mathbb{P}' and that ever element of \mathbb{P}' can be obtained in this manner by a suitable choice of the \alpha's

    \mathbb{P}' is the collection of all linear functionals on \mathbb{P}

    attempt:

    To prove that y is an element of \mathbb{P}' I show that y is a linear functional:

    <br />
\begin{aligned}<br />
y(x+z)=&y\left(\sum^n_{i=0}(\xi_i+\eta_i)t^i\right  )\\<br />
          =&\sum^n_{i=0}(\xi_i+\eta_i)\alpha_i\\<br />
          =&\sum^n_{i=0}\alpha_i\xi_i + \sum^n_{i=0}\alpha_i\eta_i\\<br />
          =&y(x)+y(z)<br />
\end{aligned}<br />

    <br />
\begin{aligned}<br />
y(\beta x)=&\sum^n_{i=0}\beta\xi_i\alpha_i\\<br />
               =&\beta\sum^n_{i=0}\xi_i\alpha_i\\<br />
               =&\beta y(x)<br />
\end{aligned}<br />

    Prove that every element can be obtained in this manner by a suitable choice of the \alpha's.

    I am not sure how to prove this, but perhaps I can say that if the set of \alpha's are a basis for \mathbb{P}', then every element of \mathbb{P}' can be written as \sum^n_{i=0}\xi_i\alpha_i.
    -------------------------------------------------------------------------

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    The problem with that is that the " \alphas" are sequences of complex numbers, not members of P' and so cannot be a basis.

    But it is easy to construct a basis:

    A basis for P itself is \{1, t, t^2, \cdot\cdot\cdot, t^n\}.

    Let L_i be the linear functional that assigns the value 1 to t^i and 0 to all other basis members.

    Then any member of P' can be written as L= a_0L_0+ a_1L_1+ a_2L_2+ \cdot\cdot\cdot+ a_nL_n and L(\xi_0+ \xi_1t+ \cdot\cdot\cdot+ \xi_nt^n)= a_0\xi_0+ a_1\xi_1+ \cdot\cdot\cdot+ a_n\xi_n
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1
    Helpful as always,thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Normed space and dual space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: June 5th 2011, 10:46 PM
  2. Dual Space of a Vector Space Question
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 16th 2011, 03:02 AM
  3. Dual space of a vector space.
    Posted in the Advanced Algebra Forum
    Replies: 15
    Last Post: March 6th 2011, 02:20 PM
  4. Replies: 3
    Last Post: March 23rd 2010, 07:05 PM
  5. vector space and its dual space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 26th 2009, 08:34 AM

Search Tags


/mathhelpforum @mathhelpforum