THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {X₂}, is an ortho- normal set in X, and let x be an arbitrary vector in X. Then the following statements are true. (1) y = [₂€ A(X, x₂)x, exists; that is, the series is summable; (2) the vector y mentioned in (1) belongs to [A];
THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {X₂}, is an ortho- normal set in X, and let x be an arbitrary vector in X. Then the following statements are true. (1) y = [₂€ A(X, x₂)x, exists; that is, the series is summable; (2) the vector y mentioned in (1) belongs to [A];
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {x}, is an ortho-
normal set in X, and let x be an arbitrary vector in X. Then the following statements
are true.
(1) y = [₂€ A(x, x₂)x₁ exists; that is, the series is summable;
(2) the vector y mentioned in (1) belongs to [A];
(3) x = [A] if and only if x = y (it can be written as the above series);
(4) x − y ¹ [A].
Proof (1). We note that Σ(x, x₂)x² converges, since
Σ ||(x, x₂)x₂||² = Σ |(x, x,)|² ≤ ||x||²
by the Bessel inequality [Theorem 9.3(1)].
Request
explain
Proof for (1) continues
Proof (2). It is clear that any partial sum of (x, x)x, must belong to [4]. This
implies that the limit y must belong to [4].
Request clocity proof (2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15b7304-cc73-4505-92c3-23aa2fda4f71%2Febbbc5b3-1414-48e7-8386-d700e1c48c61%2F81fq2bj_processed.png&w=3840&q=75)
Transcribed Image Text:THÉOREM 9.8. Let X be a Hilbert space, suppose that A = {x}, is an ortho-
normal set in X, and let x be an arbitrary vector in X. Then the following statements
are true.
(1) y = [₂€ A(x, x₂)x₁ exists; that is, the series is summable;
(2) the vector y mentioned in (1) belongs to [A];
(3) x = [A] if and only if x = y (it can be written as the above series);
(4) x − y ¹ [A].
Proof (1). We note that Σ(x, x₂)x² converges, since
Σ ||(x, x₂)x₂||² = Σ |(x, x,)|² ≤ ||x||²
by the Bessel inequality [Theorem 9.3(1)].
Request
explain
Proof for (1) continues
Proof (2). It is clear that any partial sum of (x, x)x, must belong to [4]. This
implies that the limit y must belong to [4].
Request clocity proof (2)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
