Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•,•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x,x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (iv) (v) (x, ay) = a(x, y) for all x, y V (x, y+z) = (x, y) + (x, z) for all (x, y) (y,x) for all x, y € V. = (p, q) = xεν. n and for all Example: Verify that each of the following mappings define a real inner product. i=1 x, y, z ε ν. (d) Let x₁,x2,...n be distinct real numbers and let (,): PnX Pn → R be defined as p(x₁)q(xi), a EF. for all p, q Pn.
Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•,•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x,x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (iv) (v) (x, ay) = a(x, y) for all x, y V (x, y+z) = (x, y) + (x, z) for all (x, y) (y,x) for all x, y € V. = (p, q) = xεν. n and for all Example: Verify that each of the following mappings define a real inner product. i=1 x, y, z ε ν. (d) Let x₁,x2,...n be distinct real numbers and let (,): PnX Pn → R be defined as p(x₁)q(xi), a EF. for all p, q Pn.
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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Transcribed Image Text:Definition 28. Let V be a vector space over F (here F= R, C). Then the function
(•₁•): V × V → F
is said to be an inner product on V if the following conditions are satisfied:
(x, x) is real
and (x,x) ≥ 0 for all
(x,x) = 0 if and only if
x € 0₂.
(i)
(ii)
(iii)
(x, ay) = a(x, y) for all
x, y V
(iv)
(x, y+z) = (x, y) + (x, z) for all
(v) (x, y) = (y,x) for all x, y V.
(p, q) =
72
xεν.
Example: Verify that each of the following mappings define a real inner product.
(d) Let x1,x2,... En be distinct real numbers and let (
P(xi)q(xi),
i=1
and for all
x,y,z € V.
a EF.
): Pn x Pn → R be defined as
for all p, q E Pn.
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