3 votes
Accepted

Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?

If $f$ and $g$ are $\mu-$ integrable (in the Bochner sense ) then they are almost separably valued an this reduces the proof to the case when $E$ is a separable Banach Space. Now $\int_A x^{*}\circ f ...
  • 9,843
2 votes

Eigenvectors spanning closed subspace in a Banach space

I'm looking for an example of a (bounded) linear operator T on a Banach space X with infinitely many eigenvalues such that $∑_{λ∈\mathbb C}\ker(T−λ)$ is closed where the sum denotes the algebraic sum ...
2 votes
Accepted

Intuitively explain why $U: X \rightarrow X$ is invertible if it is close enough to the identity operator.

If $T$ is any invertible operator and $\|T-U\|<\|T^{-1}\|^{-1}$, then $$ \|I-T^{-1}U\|=\|T^{-1}(T-U)\|\leq\|T^{-1}\|\,\|T-U\|<1, $$ and then by the result quoted in the question you haveh $T^{-1}...
2 votes

Let $E$ be a separable Banach space. Then ${E^{\star}}$ is metrizable in the weak$^{\star}$ topology $\sigma\left(E^{\star}, E\right)$

As @DavidMitra mentioned in a comment, my claim is wrong. Indeed, the author of the book has a remark just right after the proof of Theorem 3.28, i.e., Remark 20. One should emphasize again (see ...
  • 11.8k
1 vote

Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?

I fill some detail in @geetha290krm's excellent answer to better understand his/her ideas. We have $$ \varphi \left (\int_A f \mathrm d \mu\right ) = \varphi \left (\int_A g \mathrm d \mu \right) \...
  • 11.8k

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