Projection
 Let us have two different points . Prove that the set of points which in the Euclidean norm are closer to the point than to make up a halfspace. Is this true for another norm?
 Find if ,
 Find if ,
 Find if ,
 Illustrate the geometric inequality that connects , from which it follows that is a projection of the point onto a convex set of .
 For which sets does the projection of the point outside this set exist? Uniquie?
 Find if
 Find if ,
 Let be a closed set, and be a point not lying in it. Show that the projection in norm will be unique, while in norm this statement is not valid.
 Find the projection of the matrix on a set of matrices of rank . In Frobenius norm and spectral norm.
 Find a projection of the matrix on a set of symmetrical positive semidefinite matrices of . In Frobenius norm and the scalar product associated with it.
 Find the projection of point onto the set in norm. Consider the different positions of .
 Find , if .

Prove that projection is a nonexpansive operator, i.e. prove, that if is nonempty, closed and convex set, then for any