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Nik Weaver
Nik Weaver
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Let $V_0$ be ana closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V_0$ isn't closed?

A positive solution would let me answer this question.

Let $V_0$ be an infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V_0$ isn't closed?

A positive solution would let me answer this question.

Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V_0$ isn't closed?

A positive solution would let me answer this question.

Source Link
Nik Weaver
Nik Weaver
  • 42.8k
  • 3
  • 112
  • 213

Finding closed subspaces whose sum isn't closed

Let $V_0$ be an infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V_0$ isn't closed?

A positive solution would let me answer this question.