Results 1 to 1 of 1

Thread: Help understanding an intermediate result

  1. #1
    Junior Member
    Joined
    Apr 2015
    From
    Sweden
    Posts
    62
    Thanks
    5

    Help understanding an intermediate result

    Hello everyone

    I am reading a section of a textbook concerning The Galerkin Method, the setting are a separable Hilbert space with a Shauder basis $\displaystyle \left(\phi_i\right)_{i\in \mathbb{N}}$ and a matrix $\displaystyle A_n = \left(a\left(\phi_i, \phi_j \right)\right) \: 1 \leq i,j \leq n $ the Author then arrives at the following result,
    ... there exist some constant c > 0 such that

    $\displaystyle \forall \lambda \in \mathbb{R}^n \: c \left| \lambda \right| \leq \left| \left| \sum_i^n \lambda_i \phi_i \right| \right| $

    Hence

    $\displaystyle \forall \lambda \in \mathbb{R}^n \: \langle A_n \lambda , \lambda \rangle \geq \alpha c^2 \left| \lambda \right| ^2 $

    From the above, it follows that $\displaystyle A_n$ is one to one (that is, $\displaystyle ker(A_n)= \left \lbrace 0 \right \rbrace$ )
    My question is then why does it follow (or how does it follow) that the matrix $\displaystyle A_n$ is injective?

    Thanks

    P.S $\displaystyle \langle \: \cdot \: , \: \cdot \: \rangle $ is the Euclidean scalar product and $\displaystyle \left| \: \cdot \: \right|$ is the Euclidean norm
    Last edited by Krisly; Apr 17th 2018 at 01:17 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Intermediate Algebra
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Jun 19th 2013, 03:41 AM
  2. result
    Posted in the Algebra Forum
    Replies: 10
    Last Post: Oct 28th 2012, 01:44 PM
  3. How does this result come about?
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Aug 11th 2012, 12:05 AM
  4. help me with this intermediate theorem plz!!
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 4th 2009, 03:45 AM
  5. how can u get this result?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Oct 14th 2008, 12:23 PM

/mathhelpforum @mathhelpforum