g is a linear combination of f1, f2,...,fn iff g(N) = {0}
Functional Analysis
Linear Functionals
Vector Space
Suppose that Ɛ is a vector space, nЄN, and f1, f2,...,fn, and g are linear functionals on Ɛ.
Let N = ker(f1)∩ker(f2) ∩...∩ker(fn). Then g is a linear combination of f1, f2,...,fn if and only if g(N) = {0}.
Hint: Suppose that g(N) = {0}. Define ĝ: Ɛ /N →C by ĝ([x]) = g(x) for each x in Ɛ and h: Ɛ /N→Cn by
h([x]) = (f1(x), f2(x),..., fn(x)) for each x in Ɛ. Hence h is one-to-one and onto M = range (h). We have h - 1 M Ɛ /N . Let f = ĝo: h - 1: M→C. The f is a linear functional on M . Extend f to Cn.
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This solution is comprised of a detailed explanation of the normed linear spaces.
It contains step-by-step explanation for the following problem:
Suppose that Ɛ is a vector space, nЄN, and f1, f2,...,fn, and g are linear functionals on Ɛ.
Let N = ker(f1)∩ker(f2) ∩...∩ker(fn). Then g is a linear combination of f1, f2,...,fn if and only if g(N) = {0}.
Solution contains detailed step-by-step explanation.