An algebraic closed graph theorem
Gabriel Obi · Pacific Journal of Mathematics · 1978
In this work we consider the question as to when an everywhere defined closed linear map from a quadratic space H 1 into another such space H 2 is orthocontinuous.The following: result is proved: Let (H^Φi), (Hz,Φ 2 ) be quadratic spaces whose J_-closed subspaces are semi-simple.If T is an everywhere defined closed linear map on Hx into H 2 then T is orthocontinuous.1* Introduction* In [1], [2] Piziak generalized, algebraically, the geometry of Hubert space.He introduced the notion of quadratic space and with this studied sesquilinear forms in infinite dimensions.He showed that certain general results which are of pure algebra imply standard topological results in the context of Hubert space (e.g., an analogue of the Riesz representation theorem was proved for these spaces and this implies the Riesz representation theorem for Hubert spaces).Now, in Hubert space an everywhere defined linear operator is continuous iff its graph is closed.It is known that if T is an everywhere defined linear operator on a quadratic space and if T is orthocontinuous then the graph of T is _L -closed.The question is whether or not the converse of this is true.In [2] it is conjectured that this may not be true in general but that it may be true if our quadratic space is such that every JL -closed subspace is splitting.In this work we show that this conjecture is true.In fact we show that if every JL -closed subspace of our quadratic space is semi-simple then T is orthocontinuous.We also consider other cases where T is orthocontinuous but where H lf H 2 are neither both anisotropic nor are both such that their ± -closed subspaces are semi-simple.One of the implications of our results is that in the case of inner-product spaces the completeness of the spaces is not necessary for the "algebraic closed graph theorem" to hold.Thus the theorem holds for pre-Hilbert spaces.Surprising as this may seem at first, we point out that the algebraic closed graph theorem does not imply the closed graph theorem.This is because there may be no context in a quadratic space in which to discuss continuity.Even if such a context exists it is possible for an orthocontinuous map not to be continuous as Example 4 shows.