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

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In summary, the proof of Corollary 11.4.3 in Garling's book states that if a sequence of points in a metric space converges to a point w, and a continuous function is defined on the space, then the sequence of images under the function also converges to the image of w. This can be proven using the definition of convergence and the continuity of the function.
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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

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Peter said:
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

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!

## 1. What is an orthogonal projection in mathematics?

An orthogonal projection is a type of linear transformation that projects a vector onto a subspace in a way that preserves the angle between the original vector and its projection. In other words, it is a way of representing a vector in terms of its components along a given set of axes.

## 2. How is an orthogonal projection different from a regular projection?

An orthogonal projection is different from a regular projection in that it preserves the angle between the original vector and its projection, while a regular projection may change the angle. Additionally, an orthogonal projection is always a linear transformation, while a regular projection may or may not be.

## 3. What is the significance of orthogonal projections in mathematics?

Orthogonal projections have many important applications in mathematics, including in geometry, linear algebra, and functional analysis. They are also used in various fields such as computer graphics, signal processing, and statistics.

## 4. What is Proposition 11.4.3 in Garling's book about orthogonal projections?

Proposition 11.4.3 in Garling's book is a mathematical statement that describes the properties of orthogonal projections onto a subspace. It states that for any vector in a given subspace, its orthogonal projection onto that subspace is the closest vector to the original vector in terms of Euclidean distance.

## 5. How are orthogonal projections used in real-world applications?

Orthogonal projections have many practical applications, such as in computer graphics, where they are used to create 3D images from 2D projections. They are also used in signal processing to remove noise from signals, and in statistics to analyze data and make predictions.

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