2

Closed graph theorem says ' A closed operator between Banach spaces is continuous'.

Principle of uniform boundedness says ' If $V$ is Banach and $W$ is a normed space, then pointwise bounded family of continuous operators between them is uniformly bounded.'

I know the following implications:

  1. CGT $\Longrightarrow$ PUB.

  2. If we assume $V$ and $W$ to be Hilbert spaces in the statement of CGT, then we have PUB $\Longrightarrow$ CGT.

But, I don't know if there is proof of 'PUB $\Longrightarrow$ CGT' with as it is statement (i.e. without extra assumptions). I couldn't find it online.

Christoph
  • 24,912
ogirkar
  • 2,681
  • 14
  • 27
  • 1
    See this question at mathoverflow. The answer gives a proof of the open mapping theorem from the uniform boundedness theorem (which is sufficient since OMT is equivalent to CGT). Further, that answer is based on a direct proof of PUB implies CGT and gives a reference. – Rhys Steele May 07 '20 at 12:35
  • Thank you very much ! – ogirkar May 07 '20 at 15:33

0 Answers0