## Complete metric spaces are Baire spaces- a discussion of the proof.

I refer to the proof of the statement “Every complete metric space is a Baire space.” The proof of this statement, as given in “Introduction to Banach spaces and their Geometry”, by Bernard Beauzamy, is Let be a countable set of _open_ dense subsets of complete metric space . Take any open set . WeContinue reading “Complete metric spaces are Baire spaces- a discussion of the proof.”

Today I plan to write a treatise on spaces. are normed spaces over with the p-norm, or . Say we have the space over . This just means that , where . That is a norm is proved using standard arguments (including Minkowski’s argument, which is non-trivial). Now we have a metric in  spaces: .Continue reading

## Sufficient conditions for differentiability in multi-variable calculus.

We will be focusing on sufficient conditions of differentiability of . The theorem says that if and exist and are continuous at point , then is differentiable at . We have , which we know is partially differentiable with respect to and , but may not be differentiable in general. Differentiability at point in theContinue reading “Sufficient conditions for differentiability in multi-variable calculus.”

## Lonely Runner Conjecture- II

The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be , and let us prove “loneliness” for the runner with speed . As we know, the distance between and is given by the formula for and for , where stands for time. Also,Continue reading “Lonely Runner Conjecture- II”

## Of ellipses, hyperbolae and mugging

For as long as I can remember, I have had unnatural inertia in studying coordinate geometry. It seemed to be a pursuit of rote learning and regurgitating requisite formulae, which is something I detested. My refusal to “mug up” formulae cost me heavily in my engineering entrance exams, and I was rather proud of myselfContinue reading “Of ellipses, hyperbolae and mugging”

## The utility of trigonometrical substitutions

Today we will discuss the power of trigonometrical substitutions. Let us take the expression This is a math competition problem. One solution proceeds this way: let . Then as , we can write and . This is an elementary fact. But what is the reason for doing so? Now we have . Similarly, . TheContinue reading “The utility of trigonometrical substitutions”

Let be a mapping. We will prove that , with equality when is injective. Note that does not have to be closed, open, or even continuous for this to be true. It can be any mapping. Let . The mapping of in is . As for , it may overlap with , we the mappingContinue reading

## Axiom of Choice- a layman’s explanation.

Say you’re given the set , and asked to choose a number. Any number. You may choose , or anything that you feel like from the set. Now suppose you’re given a set , and you have absolutely no idea about what points contains. In this case, you can’t visualize the points in and pickContinue reading “Axiom of Choice- a layman’s explanation.”