Trace class operators and Hilbert-Schmidt operators

If X,Y are normed spaces, let B(X,Y ) be the set of all bounded linear maps X → Y . If T : X → Y is a linear map, I take it as known that T is bounded if and only if it is continuous if and only if E ⊆ X being bounded implies that T (E) ⊆ Y is bounded. I also take it as known that B(X,Y ) is a normed space with the operator norm, that if Y is a Banach space then B(X,Y ) is a Banach space, that if X is a Banach space then B(X) = B(X,X) is a Banach algebra, and that if H is a Hilbert space then B(H) is a C∗-algebra. An ideal I of a Banach algebra is an ideal of the algebra: to say that I is an ideal does not demand that I is a Banach subalgebra, i.e. does not demand that I is a closed subset of the Banach algebra. I is a ∗-ideal of a C∗-algebra if I is an ideal of the algebra and if A ∈ I implies that A∗ ∈ I. If X and Y are normed spaces, we take as known that the strong operator topology on B(X,Y ) is coarser than the norm topology on B(X,Y ), and thus if Tn → T in the operator norm, then Tn → T in the strong operator topology. If X is a normed space, M is a dense subspace of X, Y is a Banach space and T : M → Y is a bounded linear operator, then there is a unique element of B(X,Y ) whose restriction to M is equal to T , and we also denote this by T . If X is a normed space, define 〈·, ·〉 : X ×X∗ → C by 〈x, λ〉 = λ(x), x ∈ X,λ ∈ X∗.