V. L. Levin

Conditions for \(B\)-Completeness of Ultrabarrelled and Barrelled Spaces

(Presented by Academician P. S. Aleksandrov, 16 II 1962)

The note establishes necessary and sufficient conditions for \(B\)-completeness of ultrabarrelled and barrelled spaces. All spaces considered are assumed to be Hausdorff.

Definition 1 \((^{1})\). A topological linear space \(X\) is called \(B\)-complete if every continuous and nearly open linear mapping of it onto any topological linear space \(Y\) is open*.

Theorem 1. The following assertions about a topological linear space \(X\) are equivalent:

1) \(X\) is \(B\)-complete;

2) every nearly open linear mapping with closed graph of the space \(X\) onto any topological linear space \(Y\) is open;

3) every nearly continuous linear mapping*** with closed graph of any topological linear space \(Y\) into the quotient space of the space \(X\) by a closed subspace is continuous.

This theorem is essentially contained in \((^{1})\); a proof of the analogous theorem for locally convex spaces is given in \((^{6})\).

Definition 2 \((^{7})\). A topological linear space \((X,t)\) is called ultrabarrelled if the topology \(t\) majorizes any linear topology on \(X\) that possesses a fundamental system of \(t\)-closed neighborhoods of zero.

Theorem 2. The following assertions about an ultrabarrelled space \(X\) are equivalent:

1) \(X\) is \(B\)-complete;

2) every continuous linear mapping of the space \(X\) onto any ultrabarrelled space \(Y\) is open;

3) every linear mapping with closed graph of the space \(X\) onto any ultrabarrelled space \(Y\) is open;

4) every linear mapping with closed graph of any ultrabarrelled space \(Y\) into the quotient space of the space \(X\) by a closed subspace is continuous;

5) every linear mapping with closed graph of any ultrabarrelled space \(Y\) onto the quotient space of the space \(X\) by a closed subspace is continuous.

Proof. The implication \(1) \Rightarrow 4)\) follows from Theorem 1 and from the fact that a linear mapping of an ultrabarrelled space is nearly continuous. The implications \(4) \Rightarrow 5)\) and \(3) \Rightarrow 2)\) are obvious.

* A mapping is called nearly open if the closure of the image of every neighborhood is a neighborhood.

** Locally convex \(B\)-complete spaces were introduced by Pták \((^{2})\) (see also \((^{3–6})\)).

*** A mapping is called nearly continuous if the closure of the preimage of every neighborhood is a neighborhood.

Suppose that the space \(X\) satisfies condition 5), and let us show that then condition 3) also holds for it. Let \(\varphi\) be a linear mapping with closed graph from the space \(X\) onto an ultrabarrelled space \(Y\). Then \(\varphi^{-1}(0)\) is a closed subspace in \(X\); denote it by \(H\). The mapping \(\varphi\) is the composition of two mappings \(\varphi=\psi\circ h\): the canonical homomorphism \(h\) of the space \(X\) onto the quotient space \(X/H\) and a one-to-one linear mapping \(\psi\) of the space \(X/H\) onto the space \(Y\), while the graph of the mapping \(\psi\) is closed in \(X/H\times Y\). Consider the mapping \(\psi^{-1}\) of the space \(Y\) onto the space \(X/H\); its graph is closed in \(Y\times X/H\). By condition 5), \(\psi^{-1}\) is continuous, and hence \(\psi\) is open. Then \(\varphi\) is open as the composition of two open mappings. Thus, the implication \(5)\Rightarrow 3)\) is established.

To complete the proof it remains to show that condition 1) follows from condition 2). Let the space \(X\) satisfy condition 2), and let \(\varphi\) be an almost open continuous linear mapping of the space \(X\) onto a topological linear space \(Y\). We shall show that \(Y\) is ultrabarrelled. Denote by \(t_Y\) (\(t_X\)) the topology of the space \(Y\) (\(X\)). Let \(t\) be an arbitrary linear topology on \(Y\) possessing a fundamental system of \(t_Y\)-closed neighborhoods of zero. Consider on \(X\) the topology \(\varphi^{-1}(t)\), for which the neighborhoods of zero are sets of the form \(\varphi^{-1}(V)\), where \(V\) is a neighborhood of zero of the space \((Y,t)\). The sets \(\varphi^{-1}(V)\), where \(V\) runs through a fundamental system of \(t_Y\)-closed neighborhoods of zero of the space \((Y,t)\), form a fundamental system of \(t_X\)-closed neighborhoods of zero for the linear topology \(\varphi^{-1}(t)\) on \(X\). From the ultrabarrelledness of \(X\) it then follows that \(\varphi^{-1}(V)\) is a neighborhood of zero in \((X,t_X)\); denote it by \(U\). Since \(\varphi\) is almost open, the closure of the set \(\varphi(U)\) in \(Y\) is a neighborhood of zero, but

\[ \overline{\varphi(U)}=\overline{\varphi\bigl[\varphi^{-1}(V)\bigr]}=\overline{V}=V. \]

Then \(V\) is a neighborhood of zero in \((Y,t_Y)\), i.e. \(t_Y\) majorizes \(t\), and this means that \(Y\) is ultrabarrelled. From condition 2) it then follows that \(\varphi\) is open. The theorem is proved.

Remark. Theorem 2 remains valid if everywhere in its formulation the ultrabarrelled space is replaced by a locally convex ultrabarrelled space.

Theorem 3. The following assertions about a barrelled space \(X\) are equivalent:

1) \(X\) is \(B\)-complete;

2) every continuous linear mapping of the space \(X\) onto any barrelled space \(Y\) is open;

3) every linear mapping with closed graph of the space \(X\) onto any barrelled space \(Y\) is open;

4) every linear mapping with closed graph of any barrelled space \(Y\) into the quotient space of the space \(X\) by a closed subspace is continuous;

5) every linear mapping with closed graph of any barrelled space \(Y\) onto the quotient space of the space \(X\) by a closed subspace is continuous.

The proof is analogous to the proof of Theorem 2.

Definition 3. A topological linear space \(X\) is called \(B_r\)-complete if every one-to-one continuous and almost open linear mapping of it onto any topological linear space \(Y\) is open*.

* Locally convex \(B_r\)-complete spaces were introduced by Pták \((^{2})\) (see also \((^{3})\)).

Theorem 4. The following assertions about the ultrabarrelled space \(X\) are equivalent:

1) \(X\) is \(B_r\)-complete;

2) every one-to-one continuous linear mapping of the space \(X\) onto any ultrabarrelled space \(Y\) is open;

3) every one-to-one linear mapping with closed graph of the space \(X\) onto any ultrabarrelled space \(Y\) is open;

4) every linear mapping with closed graph of any ultrabarrelled space \(Y\) into the space \(X\) is continuous;

5) every linear mapping with closed graph of any ultrabarrelled space \(Y\) onto the space \(X\) is continuous.

Analogous theorems are valid for barrelled and locally convex ultrabarrelled spaces.

Received
16 II 1962

REFERENCES

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