Complex sentence: While initially assuming the signals are vectors in a vector space, to add meaning to the space, it is necessary to use a normed vector spacebecause normed vector spaces are equipped with a distance metric (the inner product and norm).
Split into simpler sentences: While initially assuming the signals are vectors in a vector space, to add meaning to the space, it is necessary to use a normed vector space instead. This is because a normed vector space is equipped with a distance metric (the inner product and norm).