closed range theorem

Let $X$ and $Y$ be Banach spaces, $T\; :\; D(T)\; \backslash to\; Y$ a closed linear operator whose domain $D(T)$ is dense in $X,$ and $T\text{'}$ the transpose of $T$. The theorem asserts that the following conditions are equivalent:

• $R(T),$ the range of $T,$ is closed in $Y.$
• $R(T\text{'}),$ the range of $T\text{'},$ is closed in $X\text{'},$ the dual of $X.$
• $R(T)\; =\; N(T\text{'})^\backslash perp\; =\; \backslash left\backslash \{\; y\; \backslash in\; Y\; :\; \backslash langle\; x^*,y\; \backslash rangle\; =\; 0\; \backslash quad\; \{\backslash text\{for\; all\}\}\backslash quad\; x^*\; \backslash in\; N(T\text{'})\; \backslash right\backslash \}.$
• $R(T\text{'})\; =\; N(T)^\backslash perp\; =\; \backslash left\backslash \{x^*\; \backslash in\; X\text{'}\; :\; \backslash langle\; x^*,y\; \backslash rangle\; =\; 0\; \backslash quad\; \{\backslash text\{for\; all\}\}\backslash quad\; y\; \backslash in\; N(T)\; \backslash right\backslash \}.$

Where $N(T)$ and $N(T\text{'})$ are the null space of $T$ and $T\text{'}$, respectively.

Note that there is always an inclusion $R(T)\backslash subseteq\; N(T\text{'})^\backslash perp$, because if $y=Tx$ and $x^*\backslash in\; N(T\text{'})$, then $\backslash langle\; x^*,y\backslash rangle\; =\; \backslash langle\; T\text{'}x^*,x\backslash rangle\; =\; 0$. Likewise, there is an inclusion $R(T\text{'})\backslash subseteq\; N(T)^\backslash perp$. So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.

Several corollaries are immediate from the theorem. For instance, a densely defined closed operator $T$ as above has $R(T)\; =\; Y$ if and only if the transpose $T\text{'}$ has a continuous inverse. Similarly, $R(T\text{'})\; =\; X\text{'}$ if and only if $T$ has a continuous inverse.

Since the graph of T is closed, the proof reduces to the case when $T\; :\; X\; \backslash to\; Y$ is a bounded operator between Banach spaces. Now, $T$ factors as $X\; \backslash overset\{p\}\backslash to\; X/\backslash operatorname\{ker\}T\; \backslash overset\{T\_0\}\backslash to\; \backslash operatorname\{im\}T\; \backslash overset\{i\}\backslash hookrightarrow\; Y$. Dually, $T\text{'}$ is

:$Y\text{'}\; \backslash to\; (\backslash operatorname\{im\}T)\text{'}\; \backslash overset\{T\_0\text{'}\}\backslash to\; (X/\backslash operatorname\{ker\}T)\text{'}\; \backslash to\; X\text{'}.$

Now, if $\backslash operatorname\{im\}T$ is closed, then it is Banach and so by the open mapping theorem, $T\_0$ is a topological isomorphism. It follows that $T\_0\text{'}$ is an isomorphism and then $\backslash operatorname\{im\}(T\text{'})\; =\; \backslash operatorname\{ker\}(T)^\{\backslash bot\}$. (More work is needed for the other implications.) $\backslash square$

{{reflist}}

• {{Banach Théorie des Opérations Linéaires}}
• {{Citation | last1=Yosida | first1=K. | title=Functional Analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=6th | series=Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 | year=1980}}.

{{Banach spaces}}

{{Functional Analysis}}

Category:Banach spaces

Category:Theorems in functional analysis