Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Hilbert space

  1. Dec 8, 2008 #1
    1. The problem statement, all variables and given/known data
    http://img168.imageshack.us/img168/5042/48390466ny3.png [Broken]


    2. Relevant equations

    A orthonormal system is if [tex] f_i \cdot f_j = 0[/tex] for all [tex]i \neq j[/tex] and if [tex]||f_i||=1[/tex]

    A sequence contained in [tex]l^{\infty}[/tex] is a bounded sequence.

    http://img99.imageshack.us/img99/1840/67874379ps9.png [Broken]


    3. The attempt at a solution

    My guess is that I have to use theorem 9.3 but I don't understand the notation. <x,e_n> is x just a number?
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Dec 8, 2008 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Yes, [itex]<x, e_n>[/itex] is just a number and so [itex]<x, e_n>\lambda_n[/itex], for each n, is just a number. Apply your theorem 9.3 with the [itex]\lambda_n[/itex] in that theorem equal to the [itex]<x, e_n>\lambda_n[/itex] here.
     
  4. Dec 8, 2008 #3

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi dirk_mec1! :smile:

    I haven't read the whole problem,

    but just answering the last sentence:

    x is a vector, just like e_n, and the inner product, <x,e_n> , is a number. :smile:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook