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# g is a linear combination of f1, f2,...,fn iff g(N) = {0}

Functional Analysis
Linear Functionals
Vector Space

Suppose that &#949; is a vector space, n&#1028;N, and f1, f2,...,fn, and g are linear functionals on &#949;.
Let N = ker(f1)&#8745;ker(f2) &#8745;...&#8745;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 &#285;: &#949; /N &#8594;C by &#285;([x]) = g(x) for each x in &#949; and h: &#949; /N&#8594;Cn by
h([x]) = (f1(x), f2(x),..., fn(x)) for each x in &#949;. Hence h is one-to-one and onto M = range (h). We have h - 1 M &#949; /N . Let f = &#285;o: h - 1: M&#8594;C. The f is a linear functional on M . Extend f to Cn.

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Functional Analysis
Linear Functionals
Vector Space

Suppose that ...

#### Solution Summary

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 &#949; is a vector space, n&#1028;N, and f1, f2,...,fn, and g are linear functionals on &#949;.
Let N = ker(f1)&#8745;ker(f2) &#8745;...&#8745;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.

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