How to prove the set of fourier multipliers is a banach algebra?

Question

Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose $F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$ is the Fourier transform given by $$(Ff)(x):=\int_\mathbb{R} f(t)\mathrm{e}^{itx}\,\mathrm{d}t\,,x\in\mathbb{R}$$ We know a function $b\in L^\infty(\mathbb{R})$ is called a multiplier if the convolution operator $T(b):=F^{-1}bF$ maps the dense subset $L^2(\mathbb{R})\cap L^p(\mathbb{R})$, of $L^p(\mathbb{R})$, ($1<p<\infty$) into itself and extends to a bounded linear operator on $L^p(\mathbb{R})$. Let K be the set of all multipliers on $L^p(\mathbb{R})$ we define a norm here, $\|b \|=\|T(b)\|_{B(L^p(\mathbb{R}))}$\, where ${B(L^p(\mathbb{R}))}$ is the set of all bounded linear operators acting on the space $L^p(\mathbb{R})$. How can I prove that K is a banach algebra?