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Example text
Notice that then d(f, f ) = 0, d(f, g) = d(g, f ) and for any three elements d(f, h) ≤ d(f, g) + d(g, h) by virtue of the triangle inequality. e. 0 = d(f, g) = f − g . e. satisfies condition 4. 40 CHAPTER 2. HILBERT SPACES AND COMPACT OPERATORS. A complex vector space V endowed with a scalar product is called a preHilbert space. Let V be a complex vector space and let · be a map which assigns to any f ∈ V a non-negative real f number such that f > 0 for all non-zero f . 4) it is called a norm. A vector space endowed with a norm is called a normed space.
We continue with the assumption that V is pre-Hilbert space. If A and B are two subsets of V , we write A ⊥ B if u ∈ A and v ∈ B ⇒ u ⊥ v, in other words if every element of A is perpendicular to every element of B. Similarly, we will write v ⊥ A if the element v is perpendicular to all elements of A. Finally, we will write A⊥ for the set of all v which satisfy v ⊥ A. Notice that A⊥ is always a linear subspace of V , for any A. Now let M be a (linear) subspace of V . Let v be some element of V , not necessarily belonging to M .
Let C be a totally ordered subset of F. Since M is compact, the intersection of any nested family of non-empty closed sets is again non-empty. e. 3). Indeed, since df (F ) = df (M) for any F ∈ C this means that for any g ∈ A, the sets {x ∈ F ||f (x) − g(x)| ≥ df ((M ))} are non-empty. They are also closed and nested, and hence have a non-empty intersection. So on the set E= F F ∈C we have f −g E ≥ df (M). 16. MACHADO’S THEOREM. e. 3). We shall call such a subset f -minimal. ] Suppose that A ⊂ CR (M) is a subalgebra which contains the constants and which is closed in the uniform topology.