Determine in each case below whether U is a subspace V. If it is, verify all conditions for U to be a subspace, and if not, state a condition that fails and give a counter-example showing that the condition fails.
There are parts (a) through (f) to this question. I am only having trouble with (e), which reads:
V is the space of all polynomials with real coefficients, viewed as functions [tex]\Re\rightarrow\Re[/tex] and U is the set of all differentiable functions [tex]\Re\rightarrow\Re[/tex]
A subspace of a vector space V is a subset U of V that has three properties:
- The zero vector of V is in U
- U is closed under vector addition
- U is closed under muliplication by scalars
The Attempt at a Solution
actually the only trouble I am having with this question is deciding whether or not U is even a subset of V. I know that if U is a subset of V, all the properties are satisfied, since the zero vector is an element of U, and from the basic properties of functions it is trivial that scalar multiplication and closure under vector addition holds.
I have been told that all differentiable functions can be written as polynomials via taylor series? if this is the case, the U is indeed a subset of V. Yet it is definitely true that all polynomials are differentiable, and thus one could say that V is in fact a subset of U. If the second statement I have suggested is true, then how could I use a counter example in this question? I suppose I'd need a function which is differentiable but can't be represented as a polynomial. Originally i thought of the exponential function and trig functions (cos and sine) as counter examples but they are analytic and thus can be represented by their respective taylor series which yield polynomials?
Any help would be much much appreciated