Final answer:
To prove that l(v, w) is infinite-dimensional, consider that each basis element of the finite-dimensional space v can be mapped to an infinite set of vectors in the infinite-dimensional space w. This results in an infinite number of independent mappings, making l(v, w) an infinite-dimensional space.
Step-by-step explanation:
The question concerns the dimensionality of a space denoted as l(v, w), where v is a nonzero finite-dimensional vector space, and w is an infinite-dimensional vector space. To prove that l(v, w) is infinite-dimensional, we must understand that a linear map space, also known as the space of linear transformations, from v to w will have a dimension equal to the product of the dimensions of the vector spaces involved. Since w is infinite-dimensional, multiplying any nonzero natural number (the dimension of v) by infinity will result in an infinite-dimensional space.
Moreover, in this space of linear mappings, every basis element of v can be mapped to an independent set of vectors in w. Because w is infinite-dimensional, there are an infinite number of such independent sets, hence l(v, w) cannot be spanned by a finite set of linear maps, confirming its infinite dimensionality.