Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn Banach theorem…) directly transferable if we exchanged every instance of norm with metric? I am asking because strictly speaking a metric is different than a … Read more

How to express log52\log_5 2 in terms of a and b (Refer to qn)

In my textbook, I came across this interesting question which I am currently struggling to solve: If log62=a and log53=b, express log52 in terms of a and b The solution given is ab1−a but I do not know the working behind this. What is it? Answer An alternative, although I agree with the exponential approach, … Read more

How to train for olympiads?

I’ve been training a lot for mathematical olympiads the past months, and I would like how to go further. Does anyone know interactive websites training your math skills? Or anything else that could train my math skills, but I’d like not just to do exercises from a book… Answer Here are two important websites, apart … Read more

How do you study mathematics? [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 4 years ago. Improve this question Which is (or was) your method to study pure maths in the university? I’m interested in knowing … Read more

What are amazing results concerning Euler totient function? [closed]

Closed. This question needs to be more focused. It is not currently accepting answers. Want to improve this question? Update the question so it focuses on one problem only by editing this post. Closed 3 years ago. Improve this question What are important or amazing results involving the Euler totient function? I am aware of … Read more

Why does math work if there are too many paradoxes?

I’m a newbie that studies applied maths and I have been learning about measure theory for the past few weeks and I came across things like Banach–Tarski paradox, Gödel’s incompleteness theorems, axiom of choice.. Which made me feel super uncomfortable and confused so my question is why is maths seem to be working if the … Read more

When I not seen a formula of prostapheresis in an identity

We suppose that I must to proof this identity \sin(220°)+\cos(10°)=\cos(70°) Easily if I put \cos(10°) to RHS of the identity, I can apply the formula of prostapheresis \cos(\alpha)-\cos(\beta) and I solved all. But if we not will put the \cos(10°) to the RHS, the identity is it possible to solve with a calculator or is … Read more

How Do Mathematicians ‘make up’ or perhaps more correctly ‘derive’ equations

I know this is definitely a thing otherwise we wouldn’t have equations. What are the methods for making up an equation from scratch and from making up an equation from an existing equation? The only way I have been able to do this is using dimensional analysis and I’d like to learn other ways, if … Read more