Final answer:
A closed linear subspace L of a K-Banach space V is also a Banach space because it inherits the property of completeness from V, ensuring every Cauchy sequence in L converges to an element within L itself.
Step-by-step explanation:
If we consider (V,\|\cdot\| V) as a K-Banach space and L as a closed linear subspace of V, then the subspace (L,\|\cdot\| L ) is also a Banach space. The reason for this is that a closed subspace of a Banach space inherits the completeness from the whole space. Completeness refers to the property that every Cauchy sequence in the space converges to an element within the space.
To illustrate this, consider a Cauchy sequence {x_n} in L. Since L is a subset of V and V is complete, the sequence {x_n} must converge to some element x in V. However, because L is closed, the limit x of the sequence within V actually lies in L. For a Banach space, not only must the sequence converge, but its limit must also lie within the space. Since this is true for any Cauchy sequence in L, we can conclude that (L,\|\cdot\| L ) is indeed a Banach space as well.