The approximation problem (Grothendieck 1955) asks whether every compact operator 
from a Banach space 
to a Banach space 
is the limit, in the operator
norm, of a sequence of finite-rank linear
operators. This question was answered in the negative by Enflo (1973), who provided
a deep counterexample to this problem.
Approximation Problem
See also
Approximation Property, Banach Space, Compact OperatorExplore with Wolfram|Alpha
References
Enflo, P. "A Counterexample to the Approximation Problem in Banach Spaces." Acta Math. 130, 309-317, 1973.Grothendieck, A. Produits tensoriels topologiques et espaces nucléaires. Providence, RI: Amer. Math. Soc., 1955.Referenced on Wolfram|Alpha
Approximation ProblemCite this as:
Weisstein, Eric W. "Approximation Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ApproximationProblem.html