Hausdorff Example 3: Function Spaces

Point Set Topology: For a third example, we consider function spaces. We begin with the space of continuous functions on [0,1]. As a metric space, this example is Hausdorff, but not complete. We consider Cauchy sequences and a possible completion.

[Godisahomo]

Interesting vid. I like how you mixed real analysis and point set topology into one vid.

[MathDoctorBob]

@nahaymath You're welcome! Whenever I have to think about function spaces, I alway ask myself if things makes sense for Fourier series. If Hilbert spaces, we're one step removed from linear algebra. In fact, the main reason I use L^1 instead of L^2 here is to keep the triangle inequality simple. - Bob

[Gauss]

Very clear and concise. Thank you! You rock!

[MathDoctorBob]

@nahaymath That comes in at the end. I set things up this way to keep the difficulty down as long as possible. - Bob

[MathDoctorBob]

@Godisahomo Thanks! In grad school, all the boundaries between subjects begin to disappear. It can be disorienting; a good stable of examples and counterexamples is a must. - Bob

[Vercongent]

@14:33 f~g iff f = g *except* on a set of measure zero.