Problem 1. We denote by (2 the set of all sequences u = (uk)k=1,2,. = (u1, u2, ...) (uk EC) satisfying luk| < oo. We define the inner product of as follows: k=1 (u, v) = uTE (u, v e l). k=1 Then, e is a Hilbert space. From the following choices, please choose orthonormal systems of (e. (1) {(1,0,0,0, ..), (0, 1, 0,0, ...), (0, 0, 1,0,.)}. (2) {(1,0,0,0, 0, ..), (0, 2,0, 0, 0, .), (0,0, 3, 0,0, ..), (0,0,0, 4, 0, ...),.}. 1 0,0,0,0,. /2' 1 0, V2 1 =,0,0,0, V2 1 0,0, --). 1 (3) , 0,0, .... (0,0.0 1 1 0, (4) {( 1 ,0,0,0,0, .. 1 ,0,0,0,0, .... 2 V2 1 1 1 ,0,0,... %3B 1 0,0,... V2' V2 -) (a, 1 ,0, | /2
Problem 1. We denote by (2 the set of all sequences u = (uk)k=1,2,. = (u1, u2, ...) (uk EC) satisfying luk| < oo. We define the inner product of as follows: k=1 (u, v) = uTE (u, v e l). k=1 Then, e is a Hilbert space. From the following choices, please choose orthonormal systems of (e. (1) {(1,0,0,0, ..), (0, 1, 0,0, ...), (0, 0, 1,0,.)}. (2) {(1,0,0,0, 0, ..), (0, 2,0, 0, 0, .), (0,0, 3, 0,0, ..), (0,0,0, 4, 0, ...),.}. 1 0,0,0,0,. /2' 1 0, V2 1 =,0,0,0, V2 1 0,0, --). 1 (3) , 0,0, .... (0,0.0 1 1 0, (4) {( 1 ,0,0,0,0, .. 1 ,0,0,0,0, .... 2 V2 1 1 1 ,0,0,... %3B 1 0,0,... V2' V2 -) (a, 1 ,0, | /2
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:Problem 1. We denote by (2 the set of all sequences
u = (uk)k=1,2,. = (u1, u2, ...) (uk EC)
satisfying Juk| < oo. We define the inner product of ( as follows:
k=1
(u, v) = EukTE (u, v E e).
k=1
Then, e is a Hilbert space. From the following choices, please choose orthonormal systems
of (e.
(1) {(1,0,0,0, ..), (0, 1, 0,0, ...), (0, 0, 1,0, .)}.
(2) {(1,0,0,0, 0, ..), (0, 2,0, 0, 0, ..), (0,0, 3,0,0, ..), (0,0,0, 4, 0, ...),.}.
1
0,0,0,0,.
1
0,
V2
1
,0,0,0,
V2
1
0,0,
1
(3)
, 0,0, ..),
....
(000
(4) {(
(na,
1
1
0,
v2' V2
1
,0,0,0,0, ..
V2
1
1
,0,0,0,0,....
2 V2'
1
0,0,...
%3B
/2
--) (a,
1
,0,
V2' V2
,0,0, ..
/2
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