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It is common wisdom that Hilbert space is too "small" for quantum mechanics,in that it fails to contain useful objects such as a plane wave or a δ function.Thus it cannot cope with the familiar Dirac bra- and ket formalism.A first way out is to use instead a Rigged Hilbert Space,e.g.Schwartzs tempered distributions S (C) L2(C)S’.The price to pay is to introduce sophisticated locally convex spaces.However,the triplet may be advantageously replaced by a scale of Hilbert spaces interpolating between S and S (the Hermite representation).In fact,many such families of function spaces play a central role in analysis,such as Lp spaces,Besov spaces,amalgam spaces or modulation spaces.In all such cases,the parameter indexing the family measures the behavior (regularity,decay properties) of particular functions or operators.Actually all these space families are,or contain,scales or lattices of Banach spaces,which are special cases of partial inner product spaces.So are also a single Hilbert space and a Rigged Hilbert Space.Thus partial inner product spaces offer a unifying framework for quantum mechanics.In this lecture,we shall give an overview of partial inner product spaces and operators on them,defined globally on the whole family,instead of individual spaces.We will discuss a number of operator classes,such as morphisms,pro jections or group representations and quote a number of applications in quantum mechanics.