Great question! What you need is Sandy Grabiner's approximation lemma:

Lemma: Let $E$ and $F$ be Banach spaces, let $T \in B(F,E)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [E]_1$ there exists $x_0 \in [F]_M$ with $\|y - Tx_0\| \leq r$. Then for each $y \in E$ there exists $x \in F$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.

I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is: Theorem 3.35 of my book Measure Theory and Functional Analysis.

It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.

Great question! What you need is Sandy Grabiner's approximation lemma:

Lemma: Let $E$ and $F$ be Banach spaces, let $T \in B(E,F)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [F]_1$ there exists $x_0 \in [E]_M$ with $\|y - Tx_0\| \leq r$. Then for each $y \in F$ there exists $x \in E$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.

I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is: Theorem 3.35 of my book Measure Theory and Functional Analysis.

It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.

Great question! What you need is Sandy Grabiner's approximation lemma:

Lemma: Let $E$ and $F$ be Banach spaces, let $T \in B(E,F)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [F]_1$ there exists $x_0 \in [E]_M$ with $\|y - Tx_0\| \leq r$. Then for each $y \in F$ there exists $x \in E$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.

I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is Theorem 3.35 of by book Measure Theory and Functional Analysis.

It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.

Great question! What you need is Sandy Grabiner's approximation lemma:

Lemma: Let $E$ and $F$ be Banach spaces, let $T \in B(F,E)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [E]_1$ there exists $x_0 \in [F]_M$ with $\|y - Tx_0\| \leq r$. Then for each $y \in E$ there exists $x \in F$ with $\|x\| \leq \frac{M\|y\|}{1-r}$ and $Tx = y$.

I think this immediately implies that $F = E$ in your question. You can probably prove the lemma yourself very easily (hint: geometric series), but for the sake of completeness a reference is: Theorem 3.35 of my book Measure Theory and Functional Analysis.

It's a great lemma and deserves to be better known. For instance, both the open mapping theorem and Tietze's extension theorem follow easily from it.