There are compact operators that are not norm-limits of finite-rank operators

Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators.

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Solution 1:

A Banach space for which the finite rank operators are norm-dense in the compact operators is said to have the approximation property (AP). An explicit example of a Banach space without the AP is the space $B(H)$ of bounded linear operators on an infinite-dimensional Hilbert space by deep work of Szankowski.

Banach asked in his book of 1932 whether there are examples of Banach spaces without the AP (in modern terminology). Grothendieck studied the question intensely in the fifties, trying to prove that every Banach space has the AP, but he failed. It remained an open question until Enflo constructed a counterexample in 1972 (for which he was awarded a goose). A lot of work has been put into investigating the property, but examples are still not easy to identify.

You can find a detailed discussion, references and examples in Peter G. Casazza's survey Approximation properties, Chapter 7 of the Handbook of the geometry of Banach spaces. See especially the second half of section 2.