Prove Theorem 2.2 for the case that β is infinite, that is, $R(T) = span({T(v) : v ∈ β})$.
$span({T(v) : v ∈ β})\subset R(T)$ can be deduced trivially. How do I prove converse? For finite-dimensional vector space, it can be proved without any confusion. How do I prove it for an infinite dimensional case? I got the solution from solution manual.
My Doubt:- I don't understand why did they assume $v\in V $ has a finite linear combination of elements in $(T(\beta)$?
Solution given in solution manual
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I believe that this question is about a basis on vector space.
Consider a vector space $V$ s.t. $V$ is set of all polynomial over $(0,1)$. If we allow the infinite sum, then $\sum_{i=0}^\infty\ x^i=\frac{1}{1-x}$ is not polynomial.