About the approaches of elements on normed space and Banach and Hilbert spaces

In this thesis, the explanations of the problem of approach with subspace elements of elements in normed Banach and Hilbert spaces were discussed, studied and learned. The presence of the best approach element in the approach of the elements with the finite dimensional subspace elements has been studied. In the case of strong normed spaces, the best approach element is uniquely shown in the approach of elements with subspace elements. In normal spaces, dense linear manifolds are defined everywhere, and examples are shown in linear linear manifolds. The importance of everywhere dense linear manifolds in the approach of normed space elements with these linear manifold elements has been shown. In the Separable normed spaces, the approach of the elements was studied. In Hilbert spaces, the approach of the elements with subspace elements has been studied. Particularly in Hilbert space, the existence and uniqueness of the best approximate element in the approach of many convex cluster elements has been studied. It has been shown that the Fourier series expansion of the element is the best approximation polynomial of the given element of the Fourier polynomial.