Homework SourseIndicate whether each of the statements below are true or fa
Indicate whether each of the statements below are true or false. Let l > 0. The orthogonal set: {sin(n pi x/t)}^infinity_n=1 is complete in L^2([0, l]). Let l > 0. The orthogonal set: {sin(n pi x/l)}^infinity_n=1 is complete in L^2([-l, l]).![Indicate whether each of the statements below are true or false. Let l > 0. The orthogonal set: {sin(n pi x/t)}^infinity_n=1 is complete in L^2([0, l]). Let](/WebImages/12/indicate-whether-each-of-the-statements-below-are-true-or-fa-1010789-1761521764-0.webp "Indicate whether each of the statements below are true or false. Let l > 0. The orthogonal set: {sin(n pi x/t)}^infinity_n=1 is complete in L^2([0, l]). Let")
Solution
Both d) and e) are FALSE.
Let S denote the given set of vectors (in either case).
If S were complete, there does not exist any non-zero vector (function) which is orthogonal to all the vectos in S.
But the function 1(constant function 1) , and all the cosine functions are orthogonal to S.
Hence S is not complete
