I am struggling with the following question:
Consider the space $V$ of continuous functions on [0,1] with the 2-norm $‖f‖_2^2$=$∫_0^1|f|^2$. $V$ is an incomplete normed linear space. For a continuous function φ on [0,1], define a linear map $M_φ:V⟶V$ by $M_φ f=φf$. Show that $M_\varphi$ is bounded and find its norm.
Here is what I did to show $M_\varphi$ is bounded.
$‖φf‖_2^2$=$∫_0^1|φf|^2$ ≤$‖φ‖_∞^2$ $∫_0^1|f|^2$ =$‖φ‖_∞^2$ $‖f‖_2^2$
This shows that $‖M_φ ‖≤‖φ‖_∞$, hence $M_φ$ is bounded.
I think in order to find the operator norm, it must be shown that the opposite inequality, $‖M_φ ‖\ge‖φ‖_∞$ hold. I suspect the unit vector in $C([0,1])$ must somehow be used. Here is my rough outline:
- Let $g$ be a unit vector in $C([0,1])$
Fix $ε$>0. Find $x_0\in [0,1]$ such that $g(x_0)=1$ and $‖φ‖_∞-ε≤|φ(x_0 )|≤‖φ‖_∞$
Show that $ ‖M_φ g‖_\infty \ge ‖φ‖_∞ -ε$
- Since $ ‖M_φ g‖_2 \ge ‖M_φ g‖_\infty$, we have $‖M_φ ‖≥‖φ‖_∞$
I know there is something amiss, could you please help me understand where went wrong? Thank you so much in advance, have a wonderful day!