Let X be an inner product space, let v ∈ X, a ∈ R.Let us consider the system of equations: ‖ x ‖ = 1 , ⟨ x , v ⟩ = a .If it has a solution, then by the Schwarz inequality | a | ≤ ‖ v ‖. Is it true the converse, if | a | ≤ ‖ v ‖, then that sysem has a solution?

Question

Let   X be an inner product space, let   v  ∈  X,   a  ∈  R.Let us consider the system of equations:  ‖  x  ‖  =  1  ,  ⟨  x  ,  v  ⟩  =  a  .If it has a solution, then by the Schwarz inequality   |  a  |  ≤  ‖  v  ‖. Is it true the converse, if   |  a  |  ≤  ‖  v  ‖, then that sysem has a solution?

Paxton James · Accepted Answer

Yes it always has. Let   H be the orthogonal hyperplane to   R  v. Decompose   x  =      x    1    +  λ  v with       x    1    ∈  H. Then,       ⟨    x    ,    v    ⟩    =  a imposes   λ  =      a          ‖      v              ‖        2            . With this choice of   λ, we have  ‖  x      ‖    2    =            a      2              ‖      v              ‖        2              +  ‖      x    1        ‖    2    .To conclude, it suffices to choose       x    1    ∈  H of norm       1    −                  a        2                    ‖        v                  ‖          2                    Edit : Special cases :- if   v  =  0, then   a  =  0 and there is still a solution.- If   H = {0}, e.g.   X  =  R  v, then there is not a solution if   |  a  |  &amp;lt;  ‖  v  ‖.

lobht98 · Accepted Answer

The answer is no. The problem is if   d  i  m  (  X  )  =  1 and   v  ≠  0 such that   |  a  |  &amp;lt;  |  |  v  |  |

Let X be an inner product space, let v ∈ X , a ∈<!--

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