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Let $B$ be a Banach space, $H,K$ be closed subspaces and let $K$ be finite dimensional.

Suppose $B = H\oplus K$ and $T:B\to B$ is a bounded linear operator.

How do I show that $T(B)$ is closed $\iff$ $T(H)$ is closed ?

1 Answers 1

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The internal sum of a closed subspace and a finite-dimensional subspace is closed, so if $T(H)$ is closed then $T(B) = T(H) + T(K)$ is closed. (See the solution to exercise 41 here or the comments below for a proof).

For the other direction, we can assume that $T$ is injective, otherwise we consider the factorization of $T$ over $B/\ker T = (H / (H \cap \ker T)) \oplus (K / (K \cap \ker T))$ (noting that $T$ and its factorization have the same image).

Suppose that $T(B)$ is closed. Then $T(H)$ is a subspace of $T(B)$ of finite codimension, so it has an algebraic complement $Z$, so $T(B) = T(H) \oplus Z$ as vector spaces. Since $Z$ is finite-dimensional, $Z$ is closed, so the linear operator $S \colon H \oplus Z \to T(B)$ defined by $S(h,z) = T(h) + z$ is a continuous linear bijection between Banach spaces, hence it is a homeomorphism by the open mapping theorem. Since $H$ is closed in $H \oplus Z$, we have that $T(H) = S(H,0)$ is closed in $T(B)$.

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    "The internal sum of a closed subspace and a finite-dimensional subspace is closed" How to prove this?2013-01-09
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    Let $C$ be a closed subspace and let $F$ be a finite-dimensional subspace. Consider the quotient map $\pi\colon X \to X/C$. Then $X/C$ is a normed space since $C$ is closed and the projection $\pi \colon X \to X/C$ is continuous. Since $\pi(F) \subset X/C$ is finite-dimensional, it is closed. It follows that $C + F = \pi^{-1}(\pi(F))$ is closed as a pre-image of a closed set under a continuous map.2013-01-10
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    Incidentally, if you had bothered to Google the exact sentence of your follow-up question, you'd have found [this exercise sheet](http://math.berkeley.edu/~sarason/Class_Webpages/solutions_202B_assign11.pdf) containing a full solution as a first hit...2013-01-10
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    Very good! You're welcome. // I would recommend to be a bit more patient before you downvote an answer in which there are some details you would like to have explained further. For instance, you can check if someone has visited the site since you requested information by clicking on their user name and visiting their profile. You could have seen that I wasn't given a chance to answer your query (remember people here are from all over the world!). Many people react rather negatively to downvotes and, after all, you are asking for help... Anyway, I'm glad this has been cleared up.2013-01-11
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    I'm sorry that my English isn't very good. I am not sure if I understand the comment above... At first I downvote your answer because I can't accept your answer(actually the first part I have done two months ago, but recently I forgot the way... ) I'm not sure if I expressed my idea clearly.2013-01-11
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    I said: please do not downvote without having a good reason (I can't see one in your last comment, as far as I understand it). You posted a comment with a question and you downvoted a few hours later. I suggest to wait a few days next time. Check in the user profile whether the person was here, at least once. You might get unpleasant reactions for unwarranted downvotes. People might be less ready to explain.2013-01-11
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    Ok.. I understand this situation. Yesterday I realize that maybe that time you are not here(I haven't thought about time difference). I'm sorry for my reacting that time.2013-01-12