A question about the meaning from following excerpt from german wikipedia adressing interesting feature of nuclear spaces opposing them from Banach spaces (transl.):

While normed spaces, especially Banach or Hilbert spaces, are generalizations of finite-dimensional vector spaces (over $\Bbb R$ or $\Bbb C$) maintaining the existence of norm but losing compactness properties, the focus in the case of nuclear spaces, which cannot be normalized in the infinite-dimensional case, is on the compactness properties.

Could it be made precise in which sense nuclear spaces can be regarded as those spaces maintaining the compactness properties (contrasting their "nature" radically from Banach spaces)?

A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.):

While normed spaces, especially Banach or Hilbert spaces, are generalizations of finite-dimensional vector spaces (over $\Bbb R$ or $\Bbb C$) maintaining the existence of norm but losing compactness properties, the focus in the case of nuclear spaces, which cannot be normalized in the infinite-dimensional case, is on the compactness properties.

Could it be made precise in which sense nuclear spaces can be regarded as those spaces maintaining the compactness properties (contrasting their "nature" radically from Banach spaces)?

A question about the meaning from following excerpt from german wikipedia adressing interesting feature of nuclear spaces opposing them from Banach spaces (transl.):

While normed spaces, especially Banach or Hilbert spaces, are generalizations of finite-dimensional vector spaces (over $\Bbb R$ or $\Bbb C$) maintaining the existence of norm but losing compactness properties, the focus in the case of nuclear spaces, which cannot be normalized in the infinite-dimensional case, is on the compactness properties.

Could it be made precise in which sense nuclear spaces can be regarded as those spaces maintaining the compactness properties?

A question about the meaning from following excerpt from german wikipedia adressing interesting feature of nuclear spaces opposing them from Banach spaces (transl.):

While normed spaces, especially Banach or Hilbert spaces, are generalizations of finite-dimensional vector spaces (over $\Bbb R$ or $\Bbb C$) maintaining the existence of norm but losing compactness properties, the focus in the case of nuclear spaces, which cannot be normalized in the infinite-dimensional case, is on the compactness properties.

Could it be made precise in which sense nuclear spaces can be regarded as those spaces maintaining the compactness properties (contrasting their "nature" radically from Banach spaces)?