What is an infinite-dimensional vector space and the theorems or axioms that define an infinite-dimensional vector space?
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What is an infinite-dimensional
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A vector space V over field F is said to be infinite dimensional vector dimension, if there are infinitely many linearly independent vectors in the basis of V. That, the vector space V is not spanned by a finite set of vectors.
For example: Set of polynomials, , of any degree, with real coefficients, is infinite dimensional.
Proof: Suppose, on contrary, that is finite dimensional, of dimension n. Then can be spanned by a finite set . All these polynomials are of finite expression, so there must exist a polynomial among them, having maximum degree, say m. Without loss of generality, we can say that .
Now, we can express as .
Since contains polynomial of any degree, suppose , such that .
Then , because the maximum degree of polynomial that can be obtained by the span of these polynomials is m, and deg(q(x)) = m+1.
Hence our assumption that , is finite dimensional is wrong. is an infinite dimensional vector space.
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