Consider the space $V$ of continuous functions on [0,1] with the 2-norm $\|f\|_{2}^{2}=\int_{0}^{1}|f|^{2}$. Define a liner map $M_{\varphi} : V \rightarrow V$ by $M_{\varphi} f=\varphi f$ where $ \varphi \in C([0,1])$. Is $\mathcal{A}=\left\{M_{\varphi} | \varphi \in C([0,1])\right\}$ a Banach algebra?
Since $V$ is an incomplete normed linear space, it is not enough to prove that $\mathcal{A}$ is closed. I have proved that $M_{\varphi}$ is bounded and $\|M_{\varphi}\|=\|\varphi\|_{\infty}$. How to prove that $\mathcal{A}$ is a Banach algebra? Should I use the Stone–Weierstrass theorem?
The operator norm of $M_{\phi}$ is the sup norm of $\phi$. Hence your space is isometrically isomorphic to $C[0,1]$ and this makes it a Banach space. Other properties of a Banach algebra are trivial verifications.