New Proofs on Properties of an Orthogonal Decomposition of a Hilbert Space

We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions, particularly between those which are non zero constant multiples of each other and between functions from the kernel space and the derivative image of the trace less Sobolev space.