Dimension of the quotient space $\frac{C_{0}}{M}.$

linear-algebra functional-analysis

Solution 1:

Consider the map

$$C_0 \to \mathbb R, (x_n) \mapsto x_1+x_2 + \dotsb + x_{10}.$$

It is surjective (you can always choose the first $10$ elements of the sequence as you desire, without changing the limit) and the kernel is $M$ by definition. Hence $C_0/M \cong \mathbb R$.

Related

Use $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$
A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$ [closed]
Diagonalizable Operators and characteristic polynomials
Distance between Unilateral shift and invertible operators.
Solutions in terms of the hypergeometric functions
Solve first order PDE system
Using laplace transforms to solve a piecewise defined function initial value problem
Modular reduction of fractions
What should be the limits in this integration
How do i prove how $S_5$ is generated by a two cycle and a five cycle?
Definition and example of "order of a function at a point of a curve"
Finding the square root Newton Raphson convergence proof