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- I'm reading about the (constant) rank theorem in Rudin's Principle of Mathematical Analysis (Theorem 9.32). I am stuck on a small detail in that proof concerning the continuity of linear maps with domain being a subspace of Euclidean space.
The statement of the rank theorem can be found below.
It's a bit messy, but the relevant details are that in the statement of that theorem, we have a function ##F:E\subset\mathbb R^n\to\mathbb R^m##, where ##F'(x)## has rank ##r## for every ##x\in E## (##r\leq m,r\leq n##). Now fix ##a\in E## and put ##A=F'(a)##. Let ##Y_1## be the range of ##A##.
In the proof of the theorem, Rudin first treats the case when ##r=0##. That is quite trivial. Then, when ##r>0##, he constructs a right inverse ##S:Y_1\to\mathbb R^m## of ##A##, that is, the map ##S## that satisfies ##ASy=y## for every ##y\in Y_1##. In a later claim in the proof of the theorem, we require the continuity of ##S## (to show ##A(V)## is open in ##Y_1##). Rudin has an earlier theorem where he states that linear maps from ##\mathbb R^n\to\mathbb R^m## have finite operator norm (that is, the ##\sup## of ##|Ax|## over all ##|x|\leq 1##) and are uniformly continuous (this is Theorem 9.7 in the text). I wonder, does Theorem 9.7 also apply to ##S##, whose domain ##Y_1## seems to be a subspace of ##\mathbb R^m##?
It's a bit messy, but the relevant details are that in the statement of that theorem, we have a function ##F:E\subset\mathbb R^n\to\mathbb R^m##, where ##F'(x)## has rank ##r## for every ##x\in E## (##r\leq m,r\leq n##). Now fix ##a\in E## and put ##A=F'(a)##. Let ##Y_1## be the range of ##A##.
In the proof of the theorem, Rudin first treats the case when ##r=0##. That is quite trivial. Then, when ##r>0##, he constructs a right inverse ##S:Y_1\to\mathbb R^m## of ##A##, that is, the map ##S## that satisfies ##ASy=y## for every ##y\in Y_1##. In a later claim in the proof of the theorem, we require the continuity of ##S## (to show ##A(V)## is open in ##Y_1##). Rudin has an earlier theorem where he states that linear maps from ##\mathbb R^n\to\mathbb R^m## have finite operator norm (that is, the ##\sup## of ##|Ax|## over all ##|x|\leq 1##) and are uniformly continuous (this is Theorem 9.7 in the text). I wonder, does Theorem 9.7 also apply to ##S##, whose domain ##Y_1## seems to be a subspace of ##\mathbb R^m##?