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The latter implies that a Hilbert space H is isometrically isomorphic with its dual space H ∗ . A generalization of the Riesz Representation Theorem is the Lax–Milgram Theorem (see Sect. 3). Important consequences of the Hahn–Banach Theorem are various separation theorems, such as the following one. 35 (Separation Theorem). Let X be a normed linear space, and let K ⊂ X be a closed and convex subset. If u0 ∈ X \ K, then a linear continuous functional f ∈ X ∗ and an α ∈ R exists such that f, u ≤ α for all u ∈ K, and f, u0 > α.
For ﬁxed t, let us consider the mapping v → (f (t), v). Apparently this mapping is linear, and in view of the continuous embedding V ⊂ H, we have |(f (t), v)| ≤ f (t) H v H ≤ c f (t) H v V , which shows that the mapping is bounded. , there is a b ∈ V ∗ such that b, v = (f (t), v), for all v ∈ V, where ·, · denotes the duality pairing between V and V ∗ . Next we will see that the functional b is deﬁned in a unique way. Assume there is another h ∈ H that generates the same functional b. It yields (f (t) − h, v) = 0 for all v ∈ V, and thus f (t) = h, because V ⊂ H is densely embedded.
Iii) If x ∈ X, (xn ) is a sequence in X with xn → x as n → ∞, and y ∈ A(x), then for every n ∈ N, we can ﬁnd a yn ∈ A(xn ), such that yn → y, as n → ∞. 116. For a single-valued operator A : X → Y , upper semicontinuous and lower semicontinuous in the multivalued setting is identical with continuous. For A : M → 2N having the same corresponding properties, where M and N are subsets of the Banach spaces X and Y, respectively, then M and N have to be equipped with the induced topology. Next we introduce the notion of multivalued monotone and pseudomonotone operators from a real, reﬂexive Banach space X into its dual space and formulate the main surjectivity result for these kinds of operators.
Astro-E2 Fact Sheet