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