Intereting Posts

If $b^2$ is the largest square divisor of $n$ and $a^2 \mid n$, then $a \mid b$.
Example of a an endomorphism which is not a right divisor of zero and not onto
Let $g(x)$ be analytic on every interval that does not contain $0$ is $f(t)= E$ analytic
Function whose inverse is also its derivative?
References on filter quantifiers
Find the Maclaurin series of f(x)
What are some good examples for suggestive notation?
Bounds for submartingale
product of densities
Proof of Non-Convexity
Conditional mean on uncorrelated stochastic variable
Proof of Product Rule of Limits
Hom functors and exactness
Why does the volume of a hypersphere decrease in higher dimensions?
Integral involving exponential, power and Bessel function

Here is a problem in functional analysis from Folland’s book:

If $\mathcal{M}$ is a finite-dimensional subspace of a normed vector space $\mathcal{X}$, then there is a closed subspace $\mathcal{N}$ such that $\mathcal{M}\cap \mathcal{N} = 0$ and $\mathcal{M}+\mathcal{N} = \mathcal{X}$.

I tried the following approach:

I am trying to define a projection map $\pi_{\mathcal{M}}$ from $\mathcal{X}$ to $\mathcal{M}$, which would be continuous and hence taking the inverse of any closed set would give a closed set in $\mathcal{X}$. I am confused about what the projection map would be. Please suggest some approach.

- Why are inner products in RKHS linear evaluation functionals?
- Positive bounded operators
- When are two norms equivalent on a Banach space?
- Weak convergence and weak convergence of time derivatives
- Proving $\ell^p$ is complete
- How to deduce open mapping theorem from closed graph theorem?

- On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$
- Direct sum of orthogonal subspaces
- Examples of statements that are true for real analytic functions but false for smooth functions
- An explicit construction for a “doubly weak” topology
- Example of converging subnet, when there is no converging subsequence
- Is the norm topology the same as the initial topology generated by the norm function?
- Hilbert dual space (inequality and reflexivity)
- If two norms are equivalent on a dense subspace of a normed space, are they equivalent?
- Gelfand-Naimark Theorem
- Two problems: When a countinuous bijection is a homeomorphism? Possible cardinalities of Hamel bases?

Let $\{e_1, …, e_n\}$ be a basis for $\mathcal M$. Every $x \in \mathcal M$ has then a unique representation $x = \alpha_1(x)e_1 +…+ \alpha_n(x)e_n$.

Each $\alpha_i$; is a continuous linear functional on $\mathcal M$ (a linear map from finite dimensional space is always continuous) which extends to a member of $\mathcal X^*$, by the Hahn-Banach theorem ($\mathcal X^*$ is the dual of $\mathcal X$). Let $\mathcal N$ be the intersection of the null spaces of these exten sions. Then $\mathcal X = \mathcal M\oplus \mathcal N$.

- Adapting a proof on elements of order 2: from finite groups to infinite groups
- If $n_j = p_1\cdot \ldots \cdot p_t – \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$
- Fractional Derivative Implications/Meaning?
- Why is the Euclidean metric the natural choice?
- Why are these geometric problems so hard?
- Existence in ZF of a set with countable power set
- Integral involving the hyperbolic tangent
- Proof of transitivity in Hilbert Style
- Critical values and critical points of the mapping $z\mapsto z^2 + \bar{z}$
- Does UFD imply noetherian?
- Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables
- Show that a connected graph on $n$ vertices is a tree if and only if it has $n-1$ edges.
- Representation of the dual of $C_b(X)$?
- How to diagonalize a large sparse symmetric matrix to get the eigenvalues and eigenvectors
- Area of a spherical triangle