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Without Closedness Assumption proceedings of the american mathematical society Volume 109, Number 4, August 1990 OPEN RELATION THEOREM WITHOUT CLOSEDNESS ASSUMPTION SZYMON DOLECKI (Communicated by William J. Davis) Abstract. It is shown that a convex relation such that the preimage of a bounded set has nonempty interior is lower semicontinuous throughout the al- gebraic interior of its domain. If, besides, the relation is closed-valued, then it is closed throughout the algebraic interior of its domain. 1. Introduction A well-known theorem of convex analysis says that if y is a topological vector space, /: Y —*R is convex and there exists an open set Q0 such that (1.1) SUp/ < +00, then / is continuous on the algebraic interior of its domain domf = {y: f(y) < +00} (e.g. [6]). We prove open relation and closed-graph theorems that are straightforward generalizations to convex relations of the above fact concerning convex functions. In particular, these theorems do not require graph-closedness assumptions. Let us recall that the open relation theorem has been established (as an ex- tension of the Banach open mapping [2]) by Ursescu [9] and, independently, by Robinson [7] under the provision of closed convex graph and refined by Borwein [3] who replaced closedness by convergent series closedness. (See also other developments in [4] by Borwein.) Those open relation theorems specialize in another classical fact: if / is a lower semicontinuous convex function, Y is barreled and int(dom f) / 0, then / is continuous throughout the algebraic interior of its domain (e.g. [6]). It is well known that this result may be reduced to that evoked at the very beginning by showing that int(dom /) ^ 0 implies (1.1) provided that Y is a Baire space. The relationship of our open relation theorem to the existing ones is analo- gous. Proofs of older theorems hinge on the fact that, for closed convex rela- tions, surjectivity when conjugated with the Baire property implies a bounded- Received by the editors April 10, 1989 and, in revised form, September 20, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46A55; Secondary 46A30. © 1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 1019 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1020 SZYMON DOLECK.I ness property. Our open relation theorem consists in the implication of open- ness by boundedness. But what seems to be entirely novel is that boundedness actually entails (graph-) closedness. We divide the two mentioned facts into portions that will appear as conse- quences of the results of §3. Recall that y belongs to the algebraic interior intaA of A if, for every k € Y , there exists tQ > 0 such that y + tk € A as 0 < t < t0 . 1.1. Corollary. If f: Y —*R is convex and there exists an open set Q0 such that (1.1) holds, then, for every y € intQ(dom/), there is a neighborhood Q of y such that supe / < +00. 1.2. Corollary. Iff: Y —►R is convex and there exist a neighborhood D of the origin and c0> 0 such that sup +D f < f(y0)+c0, then there is a neighborhood Q of y0, a neighborhood D of the origin and c > 0 such that, for each y € Q and every 0 < t < I, (1.2) sup f < f(y) + tc. y+tD 1.3. Corollary. If there exists an open set Q0 such that (1.1) holds, then f is upper semicontinuous throughout intQ(dom/). 1.4. Corollary. If there exists an open set Q0 such that (1.1) holds, then f is lower semicontinuous throughout intQ(dom/). 1.5. Corollary. // / is lower semicontinuous and Y is barreled, then f is continuous on int(dom/). 2. Link between functions and relations The link with the general theorems that will follow is through the epigraph of /, i.e. epif = {y, r) € Y x R : f(y) < r}. Of course, epif is a relation between Y and R, its inverse relation being the lower level relation (epi/)~r = {y.f(y)<r}. Let Y, Z be topological spaces. A relation Q c Y x Z is called lower semicontinuous at y0 if, for each open subset Q of z such that Qy0 f)Q ^0, there exists a neighborhood P of y0 for which P c Q~ Q. By definition, Q~ is open at y0 whenever Q is lower semicontinuous at y0 . It is said to be closed at y0 if, for every z ^ Qy0, there exist neighborhoods P of y0 and Q of z such that Q n ÍIP = 0 . Recall that (2.A) A function / is upper semicontinuous at yQ if and only if epif is lower semicontinuous at y0 . (2.B) A function / is lower semicontinuous at y0 if and only if epi / is closed at y0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OPEN RELATION THEOREM WITHOUT CLOSEDNESS ASSUMPTION 1021 Now let 7 be a topological vector space (2.C) A function / fulfills (1.2) if and only if there exist a ball B in R and a neighborhood D of the origin in Y such that y + tDc(epifT(f(y) + tB). (2.D) A function / fulfills (1.1) if and only if there exists a bounded subset H0 of R such that int(epi f)~H0 / 0. Finally, recall that (2.E) dom/=(epi/)"R. 3. Open and closed relation theorems Let X, Y be topological vector spaces and let A be a convex subset of XxY. 3.1. Proposition. Suppose that there is a bounded subset HQ of X such that int A//0 / 0. Then, for every y € intQ AX, there exists a bounded set H such that y eint AH. Proof. Let y0 € Y and D be a neighborhood of the origin of Y such that y0 + D c AH. Since y € intQ AX, there exists yx e AX and 0 < a < 1 such that y = ( 1 - a)y0 + ayx . Let xx € A~yx and H = ( 1 - a)HQ + axx . Then, by the convexity of A, y + (l-a)D = (l -a)(y0+D)+ayx C (1 - a)AHQ + aAxx C A[(l -a)H0 + axx]=AH, what completes the proof because H is bounded. □ 3.2. Theorem [3]. Let yQ € Ax0 and let V and W be balanced neighborhoods of 0 in Y and X, respectively, for which (3.1) yQ+WcA(x0 + V), then, for each e > 0 and balanced neighborhoods B and D of 0 in X and Y, respectively, such that V + eV c B and D + eD c W, the formula (3.2) y + tDcA(x + tB) holds for every (x, y) € (xQ+ sV) x (y0 + eD) n A and 0 < t < 1. Proof. Choose e > 0. Let D and B satisfy the assumptions. Then, for x €x0 + eV and y € y0 + eD, we have (in view of (3.1)) (3.3) y + Dcy0 + eD + Dcy0 + W cA(x0+V)cA(x + eV+V)cA(x + B). Let 0 < t < 1 . Since y € Ax , by (3.1) and by the convexity of A, we have y + tD = (1 -t)y + t(y + D) C (1 - t)Ax + tA(x + B) c A[( 1 - t)x + t(x + B)] = A(x + tB). a License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1022 SZYMON DOLECKI 3.3. Theorem. If there exists a bounded subset H0 of X such that intA//0 / 0, then A~ is lower semicontinuous on int a AX. Proof. Let y0 G intQ AX. By virtue of Proposition 3.1, there exists a bounded set H and a balanced neighborhood W of 0 such that y0 + W c A//. Let xQ € A~y0 and V be an arbitrary balanced neighborhood of the origin of X. Then there exists X > 1 such that xQ+ XV D H, that is, y0 + W c A(jc0+ AF). In view of Theorem 3.2 (applied in the special case of e = 0), we get that y0 + (1/X)W c A(*0 + F), that is A~ is lower semicontinuous at (y0, x0). G 3.4. Theorem. IfA~y0 is closed and there is a bounded subset H0 of X such that y0 € int AH0, then A~ is closed at y0. Proof. Let x <£ A~y0 . Then we may find a balanced neighborhood B of the origin in X such that (3.4) (x + 3B) n A"y0 = 0. By the convexity of A, for each k € Y and every 0 < t < 1 , (3.5) tA~ (y0 + k)+A-(yQ-tk)c(l+ t)A~ yQ. Let D be a balanced neighborhood of the origin in Y such that y0 + D c AHQ, that is, for each k € D, A~(y0 + k) n HQ ^ 0. Since HQ is bounded, there exists such a Z, 0 < r < 1, that tH0 c 5 and besides Dr G ß , so that (3.6) x + 2B c(l+t)(x + 3B). As a consequence of the former inclusion, 0 € tA~(yQ + k) + B and, in view of (3.5), for each k e D, (3.7) A-(y0-//c)c(l + 0A~y0 + 5. By (3.4) and (3.6), (1 + t)A~yQ+ B is disjoint from x + B and thus, by (3.7), A~(y0 - tk) n (x + B) = 0 for each k € D, proving the closedness of A~ at V n The boundedness assumption used in Theorems 3.2, 3.3, and 3.4 is a con- sequence of surjectivity in the case when one can apply a Baire argument.
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