If X is a real inner product space show that ‖x‖ = ‖y‖ implies that 〈x+y,x−y〉 = 0. What does this mean geometrically in X = R2? What does this condition imply if X is complex?

Math 511 Problem Set 7, due October 19 (can turn in through Oct. 22)

A couple thoughts about proving things in inner product spaces:

• The norm and inner product are intimately linked. Don’t be afraid to substitute one for the other as needed.

• Often times it’s much easier to work with the square of the norm rather than the norm itself for calculations (so you can bring in the inner product).

Now the exercises:

1. If X is a real inner product space show that ‖x‖ = ‖y‖ implies that 〈x+y,x−y〉 = 0. What does this mean geometrically in X = R2? What does this condition imply if X is complex?

2. Let X be an inner product space, y ∈ X, and (xn) a sequence in X so that xn is orthogonal to y for every n. Show that if xn → x, then x is orthogonal to y as well.

3. Show that for a sequence (xn) in an inner product space, the conditions ‖xn‖ → ‖x‖ and 〈xn,x〉→ 〈x,x〉 imply the convergence xn → x.

4. Pythagorean theorem: Let X be an inner product space, and x,y ∈ X. Prove that if 〈x,y〉 = 0, then ‖x‖2 + ‖y‖2 = ‖x + y‖2.

5. Show that in an inner product space that x is orthogonal to y if and only if ‖x + αy‖≥‖x‖ for all scalars α.

(Hint: If 〈x,y〉 6= 0, then there is some scalar α with |α| = 1 and 〈x,αy〉 < 0. Then consider the values of ‖x + αty‖, where t is a real parameter (double hint: quadratic formula?))

6. (a) Suppose that X is a real inner product space. Prove the polarization identity: for any x,y ∈ X,

〈x,y〉 = 1

4

( ‖x + y‖2 −‖x−y‖2

) (b) Suppose X is a real normed linear space such that the norm satisfies the parallelogram

equality (page 130 in Kreyszig). Prove that if we define a function 〈x,y〉 as in the polarization identity above, this function is an inner product on X so that for any x ∈ X we have 〈x,x〉 = ‖x‖2. (This shows that an inner product can be recovered from its norm.)

(c) Something to think about but not turned in: If X is a complex inner product space, the corresponding polarization identity is: for any x,y ∈ X,

〈x,y〉 = 1

4

( ‖x + y‖2 −‖x−y‖2 + i‖x + iy‖2 − i‖x− iy‖2

) .

And if a complex normed linear space obeys the parallelogram equality, its inner product can be recovered in a similar manner from the norm.

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