# A Characterization for Windowed Fourier Orthonormal Basis by Liu Y.M. By Liu Y.M.

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19 item 2 it is suﬃcient to assume F1 (x) = Y ∀x ∈ X. 20. Let X, Y, Z be topological spaces, F : X → 2Y , G : Y → 2Z be a upper semicontinuous mapping (lower semicontinuous). Then the G(ξ) mapping H = G ◦ F : X → 2Z determined by the equality H(x) = ξ∈F (x) is upper semicontinuous (lower semicontinuous). Proof. The validity of the statement immediately follows from the relation−1 −1 −1 (B) = FM G−1 (B) = F −1 G−1 (B) ∀B ∈ 2Z . 6. If F : X → K(Y ) is upper semicontinuous and G : Y → C(Z) is closed then H = G ◦ F : X → C(Z) is a closed mapping.

Really let D(Λ) yn → y strongly in X ∩ N . Since Λ is locally bounded, Λ(yn ) X ∗ ≤ K and it is possible to indicate such a subsequence {ym } that Λ(ym ) → ζ weakly in X ∗ . 1 y ∈ D(Λ), Λ(y) = ζ. It is proved in a standard way that all the sequence Λ(yn ) → Λ(y) weakly in X ∗ . However, each demicontinuous operator is radially continuous. Let X be a subspace X (for example, X = X ∩ N ). 4. The operator A : U × D(A) ⊂ X → X ∗ is called quasimonotone on X, if from U un → u ∗-weakly in U, D(A) yn → y ∈ D(A) weakly in X and lim A(un , yn ), yn − y X ≤ 0 it follows that n→∞ lim A(un , yn ), yn − ξ n→∞ X ≥ A(u, y), y − ξ ∀ξ ∈ D(A) ∩ X.

Then for any sequence xn → x0 in X, yn ∈ F (xn ) it is possible to extract the subsequence {ym } such that ym → y0 ∈ F (x0 ). Proof. } and the single-valued mapping f : X0 → X (f (2−n ) = xn , f (0) = x0 ). 2 the mapping G = F ◦ f : X0 → 2Y is upper semicontinuous. It means the mapping G0 : X0 → 2Y G0 (2−n ) = yn ) and G0 (0) = G(0) = F (x0 ) is upper semicontinuous too. } has the limit point y0 . Therefore, it is possible to choose the subsequence {ynk } converging to the point y0 . Thus, ∞ y∈ ynk ; k = m, m + 1, ...