User:Alec/Questions to do/Functional analysis/From books

Questions

1. Are the norms, $$\Vert f\Vert_\infty:\eq\mathop{\text{sup} }_{t\in[0,1]}(\vert f(t)\vert)$$ and $$\Vert f\Vert_1:\eq\int^1_0\vert f(t)\vert\mathrm{d}t$$ on $C([0,1],\mathbb{R})$ equivalentTemplate:RFACOVAOCFC?
2. Verify that the product norms are equivalent, ie $\Vert x\Vert_X+\Vert y\Vert_Y$, $\sqrt{\Vert x\Vert_X^2 + \Vert y\Vert_Y^2}$ and $\text{Max}(\Vert x\Vert_X,\ \Vert y\Vert_Y\})$ are equivalent for a product of normed spaces $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$.
3. Prove that $C([0,1],\mathbb{R})$ is an infinite dimensional vector space by exhibiting a basis (presumably Hamal Basis).