Is the following space finite-dimensional? Is the following space finite-dimensional?
$$V=\left\{f(t):[0,1]\to\mathbb{R}\mid f(0)=f'(0)=0, \int_0^1 \left(f''(u)\right)^2 \,du < \infty \right\}$$
This question is largely related to my previous question. After much thought, I tried to show the space $V$ is finite-dimensional with basis $1,t,t^2,t^3$  (even though it just does not look like one).
I considered that
$$f(t) = f(0) + f'(0)t + \int_0^t (t-u) f''(u) \,du$$
and failed to recognize the coefficients for $t^2$ and $t^3$. Any help is appreciated.

No. The functions $t^2, t^3, t^4, \dots$ all belong to $V$ and are linearly independent.
