Open Mapping Theorem

Recall that a mapping T:XYT : X \to Y, where XX and YY are topological spaces, is called open if the image under TT of each open set of XX is open in YY.

Open Mapping Theorem

A bounded linear operator between Banach spaces is open if and only if it is surjective.

Proof: Let XX and YY be Banach spaces and let T:XYT : X \to Y be a bounded linear operator. Let BXB_X and BYB_Y denote the open unit balls in XX and YY, respectively.

First, suppose that TT is surjective. The balls mBXm B_X, mNm \in \NN, cover XX. Since TT is surjective, their images mTBXmTB_X cover YY. This remains true, if we take closures: mTBX=Y\bigcup \overline{mTB_X} = Y. Hence, we have written the space YY, which is assumed to have a complete norm, as the union of countably many closed sets. It follows form the Baire Category Theorem that mTBX\overline{mTB_X} has nonempty interior for some mm. Thus there are qYq \in Y and α>0\alpha > 0 such that q+αBYmTBXq + \alpha B_Y \subset \overline{mTB_X}. Choose a pXp \in X with Tp=qTp=q. It is a well known fact, that in a normed space the translation by a vector and the multiplication with a nonzero scalar are homeomorphisms and thus compatible with taking the closure. We conclude αBYT(mBXq)\alpha B_Y \subset \overline{T(mB_X-q)}. Since mBXqmB_X-q is a bounded set, it is contained in a ball βBX\beta B_X for some β>0\beta > 0. Thus, αBYTβBX=βTBX\alpha B_Y \subset \overline{T \beta B_X} = \beta \overline{TB_X}. With γ:=α/β>0\gamma := \alpha / \beta > 0 we obtain γBYTBX\gamma B_Y \subset \overline{TB_X}.

Clearly, every yγBYy \in \gamma B_Y is the limit of a sequence (Txn)(Tx_n), where xnBXx_n \in B_X. However, the sequence (xn)(x_n) may not converge! We show that it is possible to find a convergent sequence (sn)(s_n) in 4BX4B_X such that TsnyTs_n \to y. To construct (sn)(s_n), we recursively define a sequence (yk)(y_k) with yk2kγBYy_k \in 2^{-k} \gamma B_Y for kN0k \in \NN_0. The sequence starts with y0:=y20γBYy_0 := y \in 2^0 \gamma B_Y. Given yk2kγBYy_k \in 2^{-k} \gamma B_Y, one has ykT2kBXy_k \in \overline{T 2^{-k} B_X}. By the definition of closure, there exists a xk2kBXx_k \in 2^{-k} B_X such that TxkTx_k lies in the open 2(k+1)γ2^{-(k+1)} \gamma-ball about yky_k. This means that yk+1:=ykTxk2(k+1)γBYy_{k+1} := y_k - Tx_k \in 2^{-(k+1)}\gamma B_Y. Now define sns_n as the nn-th partial sum of the series k=0xk\sum_{k=0}^{\infty} x_k. The series converges, because it converges absolutely (Here we use the completeness of XX). The latter is true because xk2k=3\sum \norm{x_k} \le \sum 2^{-k} = 3. This also shows that each sns_n and the limit x:=limsnx := \lim s_n lie in 4BX4B_X. The auxiliary sequence (yn)(y_n) converges to 00 by construction. Therefore, in the limit nn \to \infty

Tsn=k=0nTxk=k=0nykyk+1=y0yn+1y0=y,Ts_n = \sum_{k=0}^{n} Tx_k = \sum_{k=0}^{n} y_k - y_{k+1} = y_0 - y_{n+1} \to y_0 = y,

as desired. It follows from the continuity of TT that TsnTxTs_n \to Tx, thus Tx=yTx = y.

In the preceding paragraph it was proven that γBY4TBX\gamma B_Y \subset 4TB_X. Hence, δBYTBX\delta B_Y \subset TB_X where δ:=γ/4\delta := \gamma/4. To show that TT is open, consider any open set UXU \subset X. If yy lies in TUTU, there exists a xUx \in U such that Tx=yTx=y. Since UU is open, there is an ϵ>0\epsilon > 0 such that x+ϵBXUx+\epsilon B_X \subset U. Applying TT, we find y+ϵTBXTUy + \epsilon TB_X \subset TU. Combine with δBYTBX\delta B_Y \subset TB_X to see y+ϵδBXTUy + \epsilon \delta B_X \subset TU. Hence, TUTU is open. This shows that TT is indeed an open mapping.

Conversely, suppose that TT is open. TODO \square\enspace

For a bijective mapping between topological spaces, to say that it is open, is equivalent to saying that its inverse is continuous. The inverse of a bijective linear map between normed spaces is automatically linear and thus continuous if and only if it is bounded. As a corollary to the Open Mapping Theorem we obtain the following:

Bounded Inverse Theorem

If a bounded linear operator between Banach spaces is bijective, then its inverse is bounded.

Also known as Inverse Mapping Theorem.