Recent content by Office_Shredder

It's just a list of names of textbooks, technically by the way the request was worded

If you subtract ##A\cap C## from the left side of everything, and ##B\cap D## on the right side, then you have reduced to the case of no intersection as long as you can show the following result: If ##B\subset A##, ##D\subset C##, ##D\approx B##, ##A\approx C##, then ##A-B \approx C- D##

A and B are matrices you know, and ##\,omega## is a number that you know, so you can plug them all in and you get like like ##Cx=0## for some matrix ##C## that you know. You might at first guess that gives you two equations and two unknown (##x## is a vector of 2 dimensions, the 0 on the right...

I find it a bit weird to suggest someone compute the derivative of cos(x) as a substep in their struggle to compute the derivative of cos(x). I don't think the suggestion is really that appropriate - either we know the derivative of cos(x) is sin(x) and we should use that fact directly, or we...

I think if a Hilbert space has a countable Schauder basis, then it has a countable Hilbert basis. You can basically just apply Grahm Schmidt on the Schauder basis and you get a set with the same span, and everything is orthonormal. I suspect the same is true (cardinality of Schauder and...

I was expressing ##\delta## as a function of ##\epsilon##. It is not true that ##\delta \leq \epsilon## in general

I don't really understand what the point of letting you use that inequality is (especially since you proved it in question 7, so it's not like they're doing you a great favor). I honestly do not know what they expect the answer to look like. I assume you're supposed to take that ratio and...

I agree that this function is uniformly continuous, but you said it "must" be uniformly continuous and it's not true that continuous functions are uniformly continuous on open intervals in general. But I do think the thing you did is correct. No need for word salad with different forms of...

If e is a function from ##(-c(d),c(d))\to \mathbb{R}## then since the domain is an open interval the endpoints aren't in the domain. Hence proving it's continuous at the endpoints makes no sense - the function isn't even defined at those points

The squeeze principle requires the smallest and largest things to become equal in the limit. 2/3 and 2 are not equal, no matter how close t and s are...

You can use the chain rule to express the derivative of e in terms of the derivative of c. This lets you put bounds on the derivative of e. Try starting with that and then see how the mean value theorem helps

What you said sounds right to me

Yes

Yes, every Hilbert space has an orthogonal basis. I think you need to assume the axiom of choice for this to be true or something.

A hamel basis can generate exactly any element with a finite sum of basis elements. A Hilbert basis just has to be able to get arbitrarily close via finite sums. So Hilbert bases can often be smaller sets I'm not sure what your doubt is so I don't know exactly how to address it...