New questions

Jim requested to post the following question:
1. What is the size or cardinality of L1[a,b], the space of the Lebesgue integrable functions on [a,b]
Also,
2. Find a polynomial f(x) with real coefficients and f(0) = 1, such that the sums of the squares of the coefficients of f(x)ⁿ and (3x² + 7x + 2)ⁿ are the same.
3. Let C be the circle radius 1, center the origin. A point P is chosen at random on the circumference of C, and another point Q is chosen at random in the interior of C. What is the probability that the rectangle with diagonal PQ, and sides parallel to the x-axis and y-axis, lies entirely inside (or on) C?
4. G is a group consisting of m real n x n matrices with the operation of matrix multiplication. The sum of the traces of the elements of G is zero. Prove that the sum of the elements of G is the zero matrix.

New questions

1. Let C[a,b] be the set real-valued continuous functions on [a,b], and Let Cst[a,b] be the set of all constants functions on [a,b]. What is the cardinality of C[a,b]/Cst[a,b]?
2. Does C[a,b] have a countable Schauder basis?

Math for fun

Our purpose in this blog is to give the opportunity for math lovers (with formal training or with no training) the opportunity to explore fun math problems. We will encourage anyone to post some interesting questions or solutions to some of the posed questions. Even though we dont want to put any restrictions on the type of questions, it will be nice if the questions could be understandable to the untrained and amateur mathematicians.

I propose the first problem: prove that a continuous real-valued function can not have an uncountable number of relative extrema.