Orthogonal Projections .... Garling, Proposition 11.4.3 .... ....

[Math Amateur]

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...

I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help to fully understand the proof of Corollary 11.4.3 ...Garling's statement and proof of Corollary 11.4.3 reads as follows:
View attachment 8975
View attachment 8976
In the last sentence of the proof Garling asserts that if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) then we have \(\displaystyle w = P_W(x)\) ...

I cannot formulate an explicit formal and rigorous proof of this statement ... can someone please help me with this ...Help will be appreciated ...

Peter

 

[Math Amateur]

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...

I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help to fully understand the proof of Corollary 11.4.3 ...Garling's statement and proof of Corollary 11.4.3 reads as follows:In the last sentence of the proof Garling asserts that if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) then we have \(\displaystyle w = P_W(x)\) ...

I cannot formulate an explicit formal and rigorous proof of this statement ... can someone please help me with this ...Help will be appreciated ...

Peter

After reflecting on my question on the post above ... the answer may be like the folowing:To show that if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) then we have \(\displaystyle w = P_W(x)\) ...Suppose \(\displaystyle x = ( x_1, \ ... \ ... , x_d )\)and \(\displaystyle w = ( w_1, \ ... \ ... , w_k, 0, 0 \ ... \ ... 0 )\) [? ... can we assume w is of this form ? ... ]Thus if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) ... ... then we have ...\(\displaystyle x = w + z\) where \(\displaystyle z = ( 0 , \ ... \ ... , 0, x_{ k + 1 }, x_{ k + 2 } \ ... \ ... x_d)\)... therefore \(\displaystyle P_W(x) = w\)
Is that correct (and rigorous) ...?

Peter

 

[soloenergy]

Hi Peter,

I haven't read Garling's book, but I can try to help with the proof of Corollary 11.4.3. From what I understand, the corollary states that if we have a sequence of points {x_j} in a metric space (X,d) that converges to a point w in X, and a continuous function f defined on X, then the sequence {f(x_j)} also converges to f(w).

To prove this, we can use the definition of convergence in a metric space. Let \epsilon > 0 be given. Then, there exists N \in \mathbb{N} such that for all n \geq N, we have d(x_n, w) < \epsilon. Since f is continuous, we know that for any \epsilon > 0, there exists \delta > 0 such that if d(x,y) < \delta, then d(f(x), f(y)) < \epsilon.

Now, for any n \geq N, we have d(x_n, w) < \epsilon. This means that for any \delta > 0, if d(x_n, w) < \delta, then d(f(x_n), f(w)) < \epsilon. But since w_j = x_j for 1 \leq j \leq k, this means that for all n \geq N, we have d(w_n, w) < \delta. Therefore, by the continuity of f, we have d(f(w_n), f(w)) < \epsilon. This shows that {f(w_n)} converges to f(w), as desired.

I hope this helps! Let me know if you have any other questions or if you need further clarification. Best of luck with your studies!