I need to prove the following statements: (a)The set $ \mathbb{F} $ of compact operators on a Hilbert space $ \mathbb{H} $ is a norm closed linear subspace of the space of all bounded linear operators on $\mathbb{H} $ . (b) If $A \in \mathbb{F} $ , and $ B $ is a bounded linear operator on $ \mathbb{H} $ , then $ AB \in \mathbb{F} $ . (c) If $ A \in \mathbb{F} $ , then $ A^{*} \in \mathbb{F} $ .
I should use the following Lemma: An operator $A$ on a Hilbert Space $ \mathbb{H} $ is compact if and only if there exists a sequence of operators $A_n$ of finite rank (that is with range space of finite dimension) which converge in norm to $A$ .
I'll be glad to receive some guidance to it... Maybe someone will be able to explain me the meaning of a norm closed linear subspace.
Thanks in advance