Results 1 to 6 of 6

Math Help - Inner Product

  1. #1
    Senior Member Sampras's Avatar
    Joined
    May 2009
    Posts
    301

    Inner Product

    Why do we define the inner product in the infinite-dimensional vector space  C([a,b], \mathbb{R}) as  \langle f,g \rangle = \int_{a}^{b} f(x)g(x) \ dx ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sampras View Post
    Why do we define the inner product in the infinite-dimensional vector space  C([a,b], \mathbb{R}) as  \langle f,g \rangle = \int_{a}^{b} f(x)g(x) \ dx ?
    It is not the inner product but a inner product on this space. However it is commonly seen as it gives us the L_2 norm, and is associated with many orthogonal function expansions, so is useful.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2009
    Posts
    113
    It is only because in the case you are considering it's considered as a (non closed) subspace of L^2([a,b]). Endowed with its natural norm topology, C([a,b],\mathbb{R} is not a Hilbert space.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jun 2009
    Posts
    113
    It seems that Captain Black arrived a little bit earlier! Sorry because of the repetition!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Sampras's Avatar
    Joined
    May 2009
    Posts
    301
    Quote Originally Posted by CaptainBlack View Post
    It is not the inner product but a inner product on this space. However it is commonly seen as it gives us the L_2 norm, and is associated with many orthogonal function expansions, so is useful.

    CB
    I wonder what would happen if we removed the condition of symmetry. In other words, we studied spaces that only had (i) bilinearity and (ii) positive definiteness.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Aug 2009
    Posts
    67
    it turns out that we do study inner products which are not symmetric. but we do not study arbitrary such products. for complex vector spaces we usually take conjugate symmetric inner products. i have not known of any other types as i guess it does not give us a rich enough structure to be useful
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 18th 2011, 04:40 AM
  2. Replies: 6
    Last Post: September 7th 2010, 09:03 PM
  3. multivariable differential for inner product(scalar product)?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 23rd 2009, 05:40 PM
  4. Replies: 4
    Last Post: September 2nd 2009, 04:07 AM
  5. Replies: 1
    Last Post: May 14th 2008, 11:31 AM

Search Tags


/mathhelpforum @mathhelpforum