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Norms, the Dual, Continuity Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms, the Dual, Continuity Table of Contents Continuity The Evaluation Map The Evaluation Map Revisited Larry Susanka Convexity Schauder Bases in a Banach The Hahn-Banach Theorem Space Mathematics Program Norms The Banach Adjoint Bellevue College June 13, 2013 Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint THE EVALUATION MAP THE EVALUATION MAP E is linear, and also one-to-one: We suppose V is a vector space and V∗ is its dual. that is, E(x) = E(y) exactly when x = y. V∗ is, itself, a vector space so it too has a dual, V∗∗ = (V∗)∗. There is an obvious collection of members of V∗∗, namely So if E is onto then it is invertible and an isomorphism. evaluation of a functional at members of V. In that case, (V∗)∗ can be identified with (i.e. it is) V. ∗ ∗ Define the evaluation map E: V ! (V ) by If V is finite dimensional, V and V∗ have the same dimension. And it follows that (V∗)∗ has the same dimension as does V. So E(x)(f ) = f (x) for each x 2 V and f 2 V∗: E must be onto and therefore an isomorphism. The infinite dimensional case is much more delicate and we will consider the extent to which we can recover this important identification in some form later. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint CONVEXITY CONVEXITY A nonempty subset S of a real vector space V is called convex if Geometrically, and in case V = R, this means that the graph of a convex function always lies on or beneath the straight line tu + (1 − t)v 2 S 8t 2 [0; 1] and u; v 2 S: connecting any two points on the graph. For this reason convex functions are also called sublinear. In other words, all points on the line segment connecting u and v are in S whenever u and v are in S. So the region above the graph of such a function is a convex If V is any real vector space we say that a function P: X ! is R subset of R2. convex provided P(tu + (1 − t)v) ≤ tP(u) + (1 − t)P(v) 8t 2 [0; 1]: Any seminorm is convex: a seminorm is the most common source of convex functions. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint THE HAHN-BANACH THEOREM Proof. Theorem If w 2 X − Y and α; β are positive and u; v 2 Y The Hahn-Banach Theorem β α β Λu + α Λv = (α + β)Λ u + v α + β α + β Real Vector Real Vector Subspace Space β α If Y ⊂ X and P: X ! R is convex ≤ (α + β) P (u − αw) + (v + βw) α + β α + β Λ 2 ∗ Λ ≤ j ≤ β P(u − αw) + α P(v + βw): and YR satisfies P Y 1 1 So [Λu − P(u − α w)] ≤ [ P(v + β w) − Λv ]: 9Ψ 2 ∗ Λ = Ψj Ψ ≤ then XR with Y and P. α β Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Proof (Cont.) Proof (Cont.) The left side does not depend on v or β, while the right is So Λ can be extended one dimension at a time while preserving independent of α and u. So there is a real number a with its relationship with P. 1 1 Let S be the set of all linear extensions of Λ to subspaces of X sup [Λu − P(u − α w)] ≤ a ≤ inf [ P(v + β w) − Λv ]: α β which are dominated by P on their domain. u2Y v2Y α>0 β>0 Partially order this set of extensions by Θ ≤ Ψ if Ψ is an L extension of Θ. Define Θ: Y Rw ! R by Θ(v + rw) = Λv + ra for each r 2 R and v 2 Y. Chains in S have upper bounds in S and we invoke Zorn’s lemma and assert that there is a maximal member Ψ of S. Considering the cases of r positive, negative or zero separately, the definition of a yields The domain of Ψ is X, else it could be extended by one dimension, contradicting maximality. Θ(v + rw) = Λv + ra ≤ Λv + P(v + rw) − Λv = P(v + rw): Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint THE HAHN-BANACH THEOREM:COMPLEX VERSION NORMS If you want to do calculus in your space you must have limits, Corollary and the easiest way to talk about limits in a vector space is through the explicit notion of distance provided through a The Hahn-Banach Theorem norm. If V is a vector space over or , a seminorm on V is a function Complex Vector Complex Vector R C If Y Subspace ⊂ X Space and P: X ! satisfies R k · k: V ! [0; 1) P(αv + βu) ≤ jαjP(v) + jβjP(u) if u; v 2 X and jαj + jβj = 1 with the property that for any number k and vectors v and w kkvk = jkj kvk and if Λ 2 Y∗ satisfies jΛj ≤ Pj C Y and k v + w k ≤ kvk + kwk 9Ψ 2 ∗ Λ = Ψj jΨj ≤ then XC with Y and P. The seminorm is called homogeneous by virtue of the first line. The second of these is called the triangle inequality. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint SEMINORMS NORMS The triangle inequality can be tweaked slightly to produce a lower limit for the norm of a sum too. If you add the condition j kvk − kwk j ≤ k v + w k ≤ kvk + kwk: kvk = 0 when and only when v = 0 A seminormed linear space, abbreviated SNLS, is a real or the seminorm is called a norm. complex vector space endowed with a seminorm. If G: V ! F is any linear functional, the map jGj: V ! [0; 1) A seminorm satisfies given as jGj(v) = jG(v)j is a seminorm, and a common source of them too. This seminorm can never be a norm unless V has k αv + βu k ≤ jαj k v k + jβj k u k if u; v 2 X and α; β 2 F: dimension 1. normed linear space NLS So a seminorm is an example (the most important example) of a A , abbreviated , is a real or complex sublinear function as found in the statement of the vector space endowed with a norm. Hahn-Banach Theorem. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint METRIC FROM A NORM CONVERGENCE The distance between vectors v and w in a SNLS V is defined by If a sequence v0; v1; v2;::: converges in a SNLS using this d(x; y) = kv − wk: pseudometric we say that the sequence converges in seminorm (or norm). The distance notion is a pseudometric on V, and is a Sometimes this is also called strong convergence, particularly particularly nice one, having the properties when we have a norm from an inner product. d(x + z; y + z) = d(x; y) 8x; y; z 2 V (translation invariance) (There is a weaker concept of convergence which is also useful there.) d(a x; a y) = jaj d(x; y) 8x; y 2 V and a 2 F (homogeneity) In case there might be confusion about the type of convergence not required of a general metric or pseudometric. involved, we might indicate intent by Whenever a notion of distance is used in an SNLS it is this vi −! w: pseudometric which will be intended. strong This pseudometric is a metric exactly when the seminorm is a norm. Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint Norms and Continuity TOC Eval Convexity H-B Norms Continuity Eval Redux Bases Adjoint NULL VECTORS FOR A SEMINORM COMPLETENESS If V is an SNLS the set N = f x 2 V j kxk = 0 g is a vector subspace of V, sometimes called the set of null vectors for the Completeness is a very important property for us. Normed seminorm (not to be confused with vectors from the nullspace linear spaces which are complete are called Banach spaces. of a linear transformation.) R and C are themselves Banach spaces, a critical fact that is In an SNLS a sequence can converge to more than one point. used often and assumed without discussion in most first-year calculus classes. In fact, if vi ! w then If S is a vector subspace of V then S is also a SNLS space, a vi ! x exactly when x = w + y for some y 2 N. subspace of V. This implies that N is a closed set.
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