Results 1 to 3 of 3

Math Help - Parallogram law

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    133

    Parallogram law

    Hello all,

    How can I show that for any  \textbf{v},\textbf{w} \in V ( V is a vector space with an inner product and associated norm) it holds that:

    \Vert \textbf{v} + \textbf{w} \Vert^{2} + \Vert \textbf{v} - \textbf{w} \Vert^{2} = 2 \left( \Vert \textbf{v} \Vert^{2} + \Vert \textbf{w} \Vert^{2} \right)

    I have done the following:

    \Vert \textbf{v} + \textbf{w} \Vert^{2}= \langle \textbf{v}+\textbf{w},\textbf{v}+\textbf{w} \rangle = \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle+\langle \textbf{v},\textbf{w} \rangle + \langle \textbf{w},\textbf{v} \rangle

    \Vert \textbf{v} - \textbf{w} \Vert^{2}=  \Vert \textbf{v} +(- \textbf{w}) \Vert^{2} = \langle \textbf{v}+(-\textbf{w}), \textbf{v}+(-\textbf{w}) \rangle = \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle-\langle \textbf{v},\textbf{w} \rangle - \langle \textbf{w},\textbf{v} \rangle

    Adding the two we get:

     \Vert \textbf{v} + \textbf{w} \Vert^{2} + \Vert \textbf{v} - \textbf{w} \Vert^{2} = \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle+\langle \textbf{v},\textbf{w} \rangle + \langle \textbf{w},\textbf{v} \rangle + \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle-\langle \textbf{v},\textbf{w} \rangle - \langle \textbf{w},\textbf{v} \rangle =  2\langle \textbf{v},\textbf{v} \rangle +  2\langle \textbf{w},\textbf{w} \rangle = 2\left ( \Vert \textbf{v}\Vert^{2} + \textbf{w}\Vert^{2}  \right)

    Is this correct ?

    Thanks.
    Last edited by surjective; March 1st 2010 at 10:27 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51
    Yes it is
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2010
    Posts
    133

    Parallelogram-law

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: August 14th 2006, 07:31 PM

Search Tags


/mathhelpforum @mathhelpforum