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1. Show that every scalar product on R² is in the form: = ar + b(r12 +.r2) + cr22. (0.1) where a.b.c are real numbers such that the matrix (1 is strictly positive definite (which means that for every (r..r2) (1. 42) 7 0 the right-hand side of (0.1) is strictly positive).

1. Show that every scalar product on R² is in the form: = ar + b(r12 +.r2) + cr22. (0.1) where a.b.c are real numbers such that the matrix (1 is strictly positive definite (which means that for every (r..r2) (1. 42) 7 0 the right-hand side of (0.1) is strictly positive).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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11 Problems for Hilbert Spaces 1. Show that every scalar product on R² is in the form: = (1.1" | !/1 (0.1) where a.b.c are real numbers such that the matrix is strictly positive definite (which means that for every (r..r2) = (y1. 42) 7 0 the right-hand side of (0.1) is strictly positive). 2. Prove that if a Banach norm satisfics the parallelogram identity. then there is a scalar product that generates this norm. that is. if V.r. y E N||.r + y || + || .r – y ||= 2( || . ||² + || ||²). then there is a scalar product on X such that V.r E V ||. ||2= (.r..r). 3. Show that the norm on (' given by j-1 can not be given by a scalar product. 1. Let Y be the two-dimensional space cquipped with the scalar product (.r1. .r2). (y1.2) I!2 + If the subspaceY is given by Y = find Y. 5. C'onsider the space of fumctions on L(0. x) square integrable with the weight p(r) = c". This means that f E L(0. x) if and only if 1/2 < x.
Transcribed Image Text:11 Problems for Hilbert Spaces 1. Show that every scalar product on R² is in the form: = (1.1" | !/1 (0.1) where a.b.c are real numbers such that the matrix is strictly positive definite (which means that for every (r..r2) = (y1. 42) 7 0 the right-hand side of (0.1) is strictly positive). 2. Prove that if a Banach norm satisfics the parallelogram identity. then there is a scalar product that generates this norm. that is. if V.r. y E N||.r + y || + || .r – y ||= 2( || . ||² + || ||²). then there is a scalar product on X such that V.r E V ||. ||2= (.r..r). 3. Show that the norm on (' given by j-1 can not be given by a scalar product. 1. Let Y be the two-dimensional space cquipped with the scalar product (.r1. .r2). (y1.2) I!2 + If the subspaceY is given by Y = find Y. 5. C'onsider the space of fumctions on L(0. x) square integrable with the weight p(r) = c". This means that f E L(0. x) if and only if 1/2 < x.
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