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单词 ProofOfClosedGraphTheorem
释义

proof of closed graph theorem


Let T:XY be a linear mapping. Denote its graph by G(T), and let p1:X×YX and p2:X×YY be the projections onto X and Y, respectively. We remark that these projections are continuousMathworldPlanetmathPlanetmath, by definition of the productPlanetmathPlanetmathPlanetmath of Banach spacesMathworldPlanetmath.

If T is boundedPlanetmathPlanetmathPlanetmathPlanetmath, then given a sequence{(xi,Txi)} in G(T) which convergesPlanetmathPlanetmath to (x,y)X×Y, we have that

xi=p1(xi,Txi)ip1(x,y)=x

and

Txi=p2(xi,Txi)ip2(x,y)=y,

by continuity of the projections.But then, since T is continuous,

Tx=limiTxi=y.

Thus (x,y)=(x,Tx)G(T), proving that G(T) is closed.

Now suppose G(T) is closed. We remark that G(T) is a vector subspace of X×Y, and being closed, it is a Banach space. Consider the operatorT~:XG(T) defined by T~x=(x,Tx). It is clear that T~ is a bijection, its inversePlanetmathPlanetmathPlanetmath being p1|G(T), the restrictionPlanetmathPlanetmath of p1 to G(T). Since p1 is continuous on X×Y, the restriction is continuous as well; and since it is also surjective, the open mapping theoremMathworldPlanetmath implies that p1|G(T) is an open mapping, so its inverse must be continuous. That is, T~ is continuous, and consequently T=p2T~ is continuous.

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