I was given the following question:
Consider the vector space $V$ the set of all real-valued continuous functions, and subspaces $U$ and $W$, the set of all exponential functions, their sums and scalar multiples, and the set of all real-valued polynomials of any degree, respectively Prove whether the sum of subspaces $U+W$ is a direct sum.
My reasoning is that it is a direct sum, since no exponential function is polynomial and vice versa, so the intersection of both sets is empty. Does this make sense?
Best Answer
If you include $e^{0x}$ as an exponential then constants belong to both $U$ and $V$. Let us assume that this is not the case.
A finite linear combinations $\sum a_i e^{a_ix}$ of exponentials can never be polynomial unless it is $0$. If some $a_i >0$ then the linear combination tends to $\infty$ faster than any polynomial. If $a_i <0$ for all $i$ then the linear combination is bounded. The only bounded polynomials are constants. Hence the sum is direct sum.