Author Archives: Ken Leung

About Ken Leung

I'm a ordinary boy who likes Science and Mathematics.

A chain of double duals

Let X be a Banach space. We know that X embeds into its double dual X** as a closed subspace isometrically via defined by and we say that X is reflexive if this embedding is a surjection. One may continue … Continue reading

Posted in Functional analysis | 4 Comments

Fuglede’s Theorem in operator theory

The following theorem of Fuglede is a classical result in the theory of normal operators. In the proof, one can appreciate how classical analysis are used in solving problems in operator theory through functional calculus.   Let be a complex … Continue reading

Posted in Analysis, Functional analysis, Operator Theory | Leave a comment

Conway base 13 function

A real-valued function f on an interval [a, b] is said to have Intermediate Value Property (IVP) if for any number y lying between f(a) and f(b), there exists some such that f(x) = y. Of course everyone learns that if f … Continue reading

Posted in Analysis | 2 Comments

An interesting result in functional analysis

Theorem. For any bounded operators A and B on a complex Hilbert space H, it is impossible that AB – BA = I, where I is the identity operator on H. To establish the theorem, we note some elementary results … Continue reading

Posted in Analysis | 8 Comments

Intersection and union of no sets

We all know how to define the intersection and union of an arbitrary family of subsets of a set A. But what if the index set I is empty, i.e. what exactly are union and intersection of “no sets”? [Answer: … Continue reading

Posted in Set Theory | 1 Comment