Normed linear Spaces - Linear Functional Analysis, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Normed Linear Spaces over C and R

1. The field F of scalars will always be C or R.

2. Definition: A linear space over the field F of scalars is a set V satisfying

a. V is closed under vector addition: For u and v in V , u + v is in V also.

b. Vector addition is commutative and associative: For all u, v and w in V ,

u + v = v + u,

(u + v) + w = u + (v + w).

c. There is a zero element (denoted 0) in V , such that v + 0 = v for all v in V .

d. For each v in V , there an additive inverse −v such that v + (−v) = 0. (Note: We usually write u − v instead of u + (−v).)

e. V is closed under scalar multiplication: For α ∈ F and u ∈ V , αu ∈ V.

f. Scalar multiplication is associative and distributive: For all α and β in F and u and w in V ,

α(β u) = (αβ )u,

(α + β)u = αu + βu,

α(u + w) = αu + αw

g. 1 v = v for all v in V .

3. Example: Rⁿ , with the usual operations, is a vector space over R.

4. Example: Cⁿ , with the usual operations, is a vector space over C.

5. Note: Instead of the previous two examples, we could have simply stated that Fⁿ , with the usual operations, is a vector space over F.

6. Example: The set C [a, b] of F-valued continuous functions defined on [a, b] is a linear space over F. (Note: Elements of C [a, b] are continuous from the right at a and from the left at b.)

7. Example: The set C^k [a, b] of F-valued k-times continuously differentiable functions defined on [a, b], is a linear space over F. (Again, the derivatives are taken from the right at a and from the left at b.)

8. Example: Let B ⊆ Rⁿ. The set L¹(B) of functions f : Rⁿ → F satisfying

                           (1)

is a linear space over F.

9. Example: Let B ⊆ Rⁿ. The set L2(B) of functions f : Rⁿ → F satisfying

                      (2)

is a linear space over F.

10. Definition: Let V be a linear space. If U ⊆ V is closed under vector addition and scalar multiplication, then U is a subspace of V . A subspace is itself a linear space.

11. Example: Let V = R³ . If U is a subspace of V , then either

a. U = R³,
b. U is a plane through the the origin,
c. U is a line through the origin,
d. U = {0}.

12. Example: Let [a, b] be a finite interval. The set C [a, b] is a subspace of L¹ [a, b].

13. Definition: A norm ║ ║ on a linear space V is a mapping from V to R satisfying

a. ║v║ ≥ 0 for all v ∈ V .

b. ║v║ = 0 if and only if v = 0.

c. ║αv║ = |α|║v║ for all α ∈ C and v ∈ V .

d. The triangle inequality: ║u + v║≤ ║u║ + ║v║ for all u and v in V .

The norm assigns to a vector a length or magnitude.

14. The distance between vectors v and w in a normed linear space V is ║v − w║. The (closed) ball about v of radius r is

B(v, r) = {w ∈ V | ║v − w║ ≤ r}.

If you replace “less than or equal to” with “less than,” you get the open ball.

15. Example: Fⁿ is a normed linear space with

║z ║ = |z | = { |z₁|² + · · · |z_n|²}1/2 .                    (3)

16. Note: There can be more than one norm on a linear space. For example

║z ║ = |z₁| + · · · + |z_n|,                             (4)

and

                        (5)

are also norms on Fⁿ .

17. Example: C [a, b] is a normed linear space with the maximum (or L^∞) norm

                         (6)

18. Example: C^k [a, b] is a normed linear space with

                      (7)

19. Example: L¹ (B) is a normed linear space with

                (8)

20. Example: L² (B) is a normed linear space with

                (9)

21. Definition: A sequence {v_k} of vectors in a normed linear space V is convergent if there is a v ∈ V such that

║v║ − v║ → 0 as k → ∞.                     (10)

We say that {v_k} converges to v and write

or

v_k → v as _k → ∞.

22. Definition: A sequence {v_k} of vectors in a normed linear space V is Cauchy convergent if

║v_m − v_n║ → 0 as m, n → ∞.                      (11)

23. Definition: A normed linear space is complete if all Cauchy convergent sequences are convergent. A complete normed linear space is called a Banach space.

24. C [a, b], C^k [a, b], L¹ (B) and L² (B) are all Banach spaces with respect to the given norms.

25. Example: Let V be the set C [0, 2] of real-valued functions with norm

                               (12)

Although V is a normed linear space, it is not a Banach space. To see this, let

for integers k ≥ 1. Clearly, f_k ∈ V . Since

the sequence {f_k} is Cauchy convergent in V . Suppose that there were a function f in V such that

It would have to be that

which is discontinuous, and hence not in V . Thus the Cauchy convergent sequence {f_k} is not convergent (in the norm on V), and V is therefore not a Banach space.

26. Why should you bother with the distinction between Banach spaces and incomplete normed linear spaces? Many equations are solved by iterative procedures: We generate a sequence {v_k} of approximate solutions, hoping it will converge to a solution v. How do you prove convergence? You don’t know if v even exists. If the v_k live in a Banach space V with norm ║ ║, it is only necessary to show that the sequence is Cauchy convergent. Then (by the definition of completeness) you are guaranteed the existence of a v ∈ V such that v_k → v as k → ∞.

27. A norm assigns a magnitude to a vector. We’d like a notion of angle as well. To this end, we introduce inner products—generalizations of the dot product on R³ .

28. Definition: An inner product on a linear space V over F is a mapping h , i from V × V to F satisfying

a. (v , v) ≥ 0 for all v ∈ V .

b. (v , v) = 0 if and only if v = 0.

c. (u , v) = (v , u)^∗ for all u and v in V .

d. (αu , v) = α(u , v) for all α ∈ F and u and v in V .

e. (u + v , w) = (u , w) + (v , w) for all u, v and w in V .

29. Note: If V is a linear space over R, then (u , v) is a real number. In this case (c) becomes

(u , v) = (v , u), for all u and v in V .

30. Example: Fⁿ is an inner product space: For x = (x₁ , . . . , x_n) and y = (y₁ , . . . , y_n), in Fⁿ ,

(x , y) = x₁ y₁^∗ + · · · + x_ny_n^∗ .                              (13)

Note that when F = R, this reduces to the usual dot product on Rⁿ :

(x , y) = x · y = x₁ y₁ + · · · + x_n y_n .                      (14)

31. Example: For a vector of positive weights w = (w₁ , . . . , w_n),

(x , y) = w₁ x₁ y₁^∗ + · · · + w_n x_n y_n^∗ ,                         (15)

is an inner product on Fⁿ .

32. Example: L² (B) is an inner product space with

                 (16)

33. Example: Let w : Rⁿ → R be bounded, real-valued and positive on B. Then for f and g taking Rⁿ to R,

                      (17)

defines an inner product.

34. Let V be an inner product space. For v ∈ V , set

                     (18)

The notation suggests that (18) defines a norm on V . We’ll show that this is the case.

35. The Cauchy-Schwarz Inequality: For all u and v in V ,

                (19)

36. It follows easily from (19) that

                   (20)

From (20) and properties (a), (b) and (d) of the inner product, we see that (18) really does define a norm. Thus an inner product space is automatically a normed linear space.

37. If the inner product space is L2 (B ), then the Cauchy-Schwarz inequality becomes

38. An inner product space has a richer geometry than a space that is merely normed. In a normed space we only have length. In an inner product space we have length and angle: We define the angle θ between u and v in an inner product space by

              (21)

This generalizes the formula for the angle between two vectors in C³ .

39. Vectors u and v in an inner product space are called orthogonal if

(u , v) = 0.

40. Definition: An inner product space that is complete with respect to the norm (18) is called a Hilbert space.

41. Cⁿ and L²(B) are Hilbert spaces with the given inner products. In a sense, there are no more (separable) Hilbert spaces. Any n-dimensional Hilbert space is an algebraic and geometric copy of Cn , and any infinite-dimensional (separable) Hilbert space is an algebraic and geometric copy of L² (B).