Orthogonal projection question

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SUMMARY

The discussion focuses on proving that the orthogonal projection \( P_K: H \rightarrow H \) in a Hilbert space \( H \) onto a nonempty, convex, closed subset \( K \) is non-expansive. The key inequality to demonstrate is \( \| P_K(x) - P_K(y) \| \leq \| x - y \| \). The user expresses confusion regarding the proof, particularly in relating the distances between points and their projections. Relevant theorems and definitions of orthogonal projections are sought for clarity.

PREREQUISITES
  • Understanding of Hilbert spaces and their properties
  • Knowledge of convex sets and their characteristics
  • Familiarity with the concept of orthogonal projections
  • Basic grasp of vector norms and inequalities
NEXT STEPS
  • Study the definition and properties of orthogonal projections in Hilbert spaces
  • Explore the proof of the non-expansive property of projections
  • Review relevant theorems such as the Projection Theorem in Hilbert spaces
  • Investigate examples of convex sets and their projections in practical applications
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Mathematicians, students studying functional analysis, and anyone interested in the properties of Hilbert spaces and orthogonal projections.

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Homework Statement



Hello,

H is a Hilbert space. K is a nonempty, convex, closed subset of H. Prove that the orthogonal projection Pk: H → H, is non-expansive:

ll Pk(x) - Pk(y) ll ≤ ll x - y ll

The Attempt at a Solution



So the length between the Pk's, which is in K (convex) is less than the distance of x and y, (x and y not in K, I assume.)

That's my thought, but I'm getting a bit confused proving it, as I'm mixing things with " ll x - Pk(x) ll ≤ ll x - y ll, " using length of vectors.

Any hint? Thanks.
 
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Can you tell us something more about the projection P_K? How is this defined for example? What theorems do you think are relevant?
 

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