Below the fold, I am giving some sample questions for the 245B midterm next week. These are drawn from my previous 245A and 245B exams (with some modifications), and the solutions can be found by searching my previous web pages for those courses. (The homework assignments are, of course, another good source of practice problems.) Note that the actual midterm questions are likely to be somewhat shorter than the ones provided here (this is particularly the case for those questions with multiple parts). More info on the midterm can be found at the class web page, of course.

(These questions are of course primarily intended for my students than for my regular blog readers; but anyone is welcome to comment if they wish.)

**Question 1.** Let be a -finite measure space, and let be a signed -finite measure. Show that the following are equivalent:

- .
- .
- and -a.e..

**Question 2.** Let be the real line with the Borel -algebra, and let be a -finite unsigned measure. Show that it is not possible for counting measure on the real line (restricted to ) to be absolutely continuous with respect to , i.e. .

**Question 3.** Let , let be the Banach space of power summable real sequences with the usual norm. For each natural number n, let be the element of consisting of the sequence which equals 1 at the entry and 0 elsewhere, thus when and otherwise. Let be a sequence in a Banach space X.

- Show that there is at most one continuous linear transformation with the property that for all natural numbers n.
- If , show that there exists a continuous linear transformation with for all natural numbers n if and only if the sequence is bounded.
- If , show that the uniqueness claim 1. can fail.

**Question 4.** Let W be a vector space, let A be an index set, and for every , let be a subspace of W which is equipped with a norm . Suppose that for each , the normed vector space is a Banach space. Assume also the following compatibility condition: if and is a sequence in which converges in norm to x and in norm to y, then x is necessarily equal to y. Show that the space

equipped with the norm

is also a Banach space.

**Question 5.** Let H be a Hilbert space, and let

be an increasing sequence of closed subspaces of H. Let be the closure of the union of these subspaces; this is another closed subspace of H. Show that for any x in H, the sequence converges in norm to , where is the orthogonal projection of x to V, and similarly for .

**Question 6.** Let H be a Hilbert space, and let V be a closed subspace of H which is non-trivial (i.e. not equal to ). Let be the orthogonal projection onto V.

- Show that has operator norm 1.
- Show that is self-adjoint, i.e. .
- Show that is idempotent, i.e. .
- Conversely, if is a self-adjoint idempotent continuous linear transformation of operator norm 1, show that for some non-trivial subspace of H.

[*Update*, Feb 2: Question 4 corrected.]

## 7 comments

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25 January, 2009 at 8:34 pm

AnonymousWhat’s this “ocunting measure” in question 2? Sounds dirty …

[Fixed – T.]30 January, 2009 at 9:48 am

AnonymousFor question 2, I also don’t understand the counting measure. So it lives on the sigma-algebra consisting of all the subsets of R, but niu is with Borel sigma-algebra. So they lives on different sigma-algebra.

30 January, 2009 at 9:57 am

Terence TaoFair enough; I intended to restrict counting measure to the Borel sigma algebra, and have now indicated this accordingly. (A measure on a sigma algebra can always be restricted to a sub-sigma algebra without difficulty.)

15 April, 2009 at 4:03 am

AnonymousDear professor!

I have a question which relate adjoint space. That is: ” Is linear isomorphism with ?” is space of continous function in . If we choose inner product of is , by then (isormophism)?

Thank you very much!

5 February, 2013 at 9:58 pm

Adam AzzamI hate to be this comment; but I suppose that should also be non-trivial (so that the operator norm of the orthogonal projection is non-zero).

[Corrected, thanks – T.]29 December, 2022 at 8:06 am

Aditya Guha RoyWow, I really like this collection of problems.

Do you also have a sample final term, or is it possible to share the actualy finals and actual mid term questions? I followed this series of lectures very closely, and would surely like to give the problems a shot.

(I’d soon post my solutions / attempts to solve these problems in a comment below.)

29 December, 2022 at 9:05 am

Aditya Guha RoySolution to Q6:

Part 1: Follows from Pythagorean theorem, and by taking a unit vector in the subspace.

Part 2: We have but the second term vanishes since the residual is orthogonal to every element of and hence in particular it is orthogonal to Similarly, one has

Part 3: This follows from definition, since is the unique element of which minimizes over all

Part 4: First of all notice that the range of a bounded idempotent operator is closed ( thus in particular is in the range of ).

Let be the range of then one has (thanks to idempotent property) for all Now we have latex \langle x , y – Py \rangle = 0$ for all but this is equivalent to saying that or in other words, that is the orthogonal projection of onto The claim follows.

Solution to Q5: We will use the fact that projections are bounded linear operators. (One way to show this is to use Part 1 of Q6.)

To begin, notice that since is a closed subspace of so Thus it suffices to show that for all

If then there is a sequence of points such that as One has (by the triangle inequality):

using uniform boundedness principle the second term on the right hand side of the line of display goes to zero as and the other quantities also go to zero as since as

(PS: The comment above this one was made without reading that the solutions are already available; so I’ll post the solutions, only if I find signficantly distinct solutions to the remaining ones’.)