## Halmos Chapter one Section 13 and 14: Linear Functionals and Bracket Notation

1.

**Problem:** Consider the set of complex numbers as a vector space over . Suppose that for each in (where ) the function is given by

**a) **

**b)**

**c) **

**d) **

**e)**

In which cases are these linear functionals?

**Proof:**

**a)** Clearly so it is indeed a map from the vector space to it’s underlying field, and so it’s a functional. To see it’s linear we merely notice that if and then

and so

so that it is indeed linear.

**b) **Similarly, we can see that so it is indeed a function and if are defined above we get again that

and so

so it is linear.

**c) **This is not linear, notice that

**d) **This is not a functional since surjectively. More simply, notice that

**e)** This is not linear. Notice that

and so is not linear.

2.

**Problem:** Suppose that for each the function is defined by

**a) **

**b)**

**c) **

**d) **

Which of these are linear functions for ?

**Proof:** Assuming we’re taking the underlying field to be …

**a) **This is a functional since clearly . Now, suppose that and then clearly

and so

so that is clearly linear.

**b) **This is not linear, since

**c)** This is not linear since

**d) **Clearly this is a functional since . Furthermore, let be as in **a)** then we see once again that

and so

which upon expansion and regrouping is

and so is linear.

3.

**Problem:** Suppose that for each define the function by

**a) **

**b) **

**c) **

**d) **

**e)**

**f)**

Which of these are linear functionals?

**Proof:**

**a) **Clearly and so it’s a functional. Furthermore, using elementary properties of integrals we see that

but clearly this is equal to

**b) **This is not linear since

**c) **This is clearly a functional since and

but this is clearly equal to

**d) **This is clearly a functional since and we see that

but clearly this is equal to

**e)** This is not a functional since

**d) **This is clearly a linear function since . Also,

but this is evidently equal to

4.

**Problem:** If is an arbitrary sequence of complex numbers, and if where write . Prove that (where the asterisk indicates the dual space) and that every element of can be obtained in this manner by a suitable choice of the ‘s

**Proof:** Clearly and we see that if and where we may assume WLOG that then

where we take . Thus,

but by how we defined the we may rewrite this as

from where it follows that .

The second part follows as an immediate corollary from the following technical lemma

**Lemma:** Let be a vector space over the field and let be a basis for . Then, any is completely determined on , in the sense that if is such that then

**Proof:** Let then by assumption there exists a unique representation

thus

but by assumption so that the above says

.

Well, actually the result isn’t really a corollary, but if are defined as above then we can see that

where it’s understood (by the unique representation of a vector in by a finite linear combination of elements of ) that for all but finitely many . In other words, for our example that sequence may be taken to be

5.

**Problem:** If for a vector space over a field is it true that is surjective?

**Proof: **Yes, since there is some such that . Then, for any we have that .

6.

**Problem: **Let be a vector space and $\varphi,\psi\in\text{Hom}\left(\mathscr{V},F\right)$ be such that

then, for some

**Proof: **

**Lemma: **Let and be such that . Then,

**Proof: **We merely note that if then

the first term clearly being in and the second in . The conclusion follows by noticing that evidently .

So, now with this we can solve the problem. Clearly if this is trivial since by assumption this would imply that . So, assume not, then there exists some such that . We claim then that

To see this let be arbitrary. Then, where . Thus,

and so the conclusion follows.

I really appreciate your work. Do you have any solutions for problem 6 from the same sections: Linear Functionals and Brackets? Thank you in advance!

Comment by popita | November 8, 2010 |

Thank you. I guess I just forgot to post it. Be sure to respond to this so that I remember to write it up, it’s really late here.

Comment by drexel28 | November 9, 2010 |

Thank you for your prompt answer! Can you also post this problem 6 from these sections? Thanks!

Comment by ralucatoscano | November 9, 2010

There you go friend.

Comment by drexel28 | November 10, 2010

I really appreciate your work! Do you have any solutions for problem 6 of Section 14: Brackets (page 22)? Thank you in advance!

Comment by ralucatoscano | November 8, 2010 |

1.I really appreciate your work! Do you have any solutions for problem 6 of Section 14: Brackets (page 22)? Thank you in advance!

Comment by ralucatoscano | November 8, 2010 |

Resp. Sir/Ma’am,

Myself Deepak, i request how can i download all proof and problems of P R Halmos form your web site.

Thanking you

Comment by Deepak Gawali | August 13, 2011 |