Chapter 4.3, Problem 46E

Calculus Let $W_1,W_2,W_3,W_4$, and $W_5$ be defined as in Example $5$. Show that $W_i$ is a subspace of $W_j$ for $i\le j$.

Example 5 Subspaces of Functions (Calculus)

Let $W_5$ be the vector space of all functions defined on $\left[0,1\right]$, and let $W_1,W_2,W_3,$ and $W_4$ be defined as shown below.

$\begin{array}{l}{W}_1=\text{set of all polynomial functions that are defined on }\left[0,1\right]\\ W_2=\text{set of all functions that are differentiable on }\left[0,1\right]\\ W_3=\text{set of all functions that are continuous on }\left[0,1\right]\\ W_4=\text{set of all functions that are integrable on }\left[0,1\right]\\ \text{Show that }{W}_1\subset W_2\subset W_3\subset W_4\subset W_5\text{ and that }{W}_i\text{ is a subspace of}{W}_j\text{for }i\le j.\end{array}$