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In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Let $X$ and $Y$ be Banach spaces, $T:D(T)\to Y$ a closed linear operator whose domain $D(T)$ is dense in $X$, and $T^{\prime }$ the transpose of $T$. The theorem asserts that the following conditions are equivalent:

• $R(T)$, the range of $T$, is closed in $Y$,
• $R(T^{\prime })$, the range of $T^{\prime }$, is closed in $X^{\prime }$, the dual of $X$,
• $R(T)=N(T^{\prime }{)}^{\perp }=\{y\in Y|⟨x^{*},y⟩=0\text{for all}{x}^{*}\in N(T^{\prime })\}$,
• $R(T^{\prime })=N(T{)}^{\perp }=\{x^{*}\in X^{\prime }|⟨x^{*},y⟩=0\text{for all}y\in N(T)\}$.

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 $T^{\prime }$ has a continuous inverse. Similarly, $R(T^{\prime })=X^{\prime }$ if and only if $T$ has a continuous inverse.

See also

• Fundamental theorem of linear algebra

References

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