Intereting Posts

Is there any explicit formula for $x_n$?
Is it true that if ${\bf{x}}^\text{T}{\bf{Ay}}={\bf{x}}^\text{T}\bf{By}$ for all $\bf{x},\bf{y}$ then $\bf{A}=\bf{B}$?
Can $f(g(x))$ be a polynomial?
Newton's method for square roots 'jumps' through the continued fraction convergents
$\sum \limits_{n=1}^{\infty}{a_n^2}$ converges $\implies \sum \limits_{n=1}^{\infty}{\dfrac{a_n}{n}}$
Prove that this ring is an integral domain based on newly defined binary operations
Krull dimension of the injective hull of residue field
Prove that $\Bbb R^2 – \{0\}$ is homeomorphic to $S^1 \times \Bbb R$.
About eigenvalues and complex matrix
Divergence of $\sum\frac{\cos(\sqrt{n}x)}{\sqrt{n}}$
Help minimizing function
Does the assertion that every two cardinalities are comparable imply the axiom of choice?
An easy question on complex
Calculating Dini derivatives for $f(x)=\begin{cases}x\,\sin{\left(\frac{1}{x}\right)} & x\neq 0\\ 0 & x=0\end{cases}$
For given $n\times n$ matrix $A$ singular matrix, prove that $\operatorname{rank}(\operatorname{adj}A) \leq 1$

How to show that $L^2([0,1])$ is a set of first category in $L^1([0,1])$?

Thank you.

- Riesz's Lemma on $l_\infty$
- Is compactness a stronger form of continuity?
- Convergence in $\mathcal{D}'(\mathbb{R})$
- Understanding definition of tensor product
- Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$
- Proving $\ell^p$ is complete

- Are continuous functions with compact support bounded?
- What is the topological dual of a dual space with the weak* topology?
- Comparing Hilbert spaces and Banach spaces.
- Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base
- If for every $v\in V$ $\langle v,v\rangle_{1} = \langle v,v \rangle_{2}$ then $\langle\cdot,\cdot \rangle_{1} = \langle\cdot,\cdot \rangle_{2}$
- Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete
- Show $T: C() \rightarrow C()$ is compact
- application of positive linear functionl
- Does Closed Graph imply Closed Range
- Understanding positive definite kernel

For $n \in \mathbb N$ set $B_n = \{f \in L^2[0,1]\mid \|f\|_2 \le n \}$. We will show that $B_n$ is nowhere dense in $L^1$. Let $g \in L^1[0,1]\setminus L^2[0,1]$ and $f \in B_n$, then $f + \frac 1k g \to f$ in $L^1$ but $f+\frac 1k g \not\in B_n$ for all $k$. Hence $f \not\in \mathring{B_n}$ and $B_n$ doesn’t have inner points. On the other hand, $B_n$ is closed in $L^1$: Let $g \in L^1$ and $g_k \in B_n$, $g_k \to g$. Then $g_{k_\ell} \to g$ almost everywhere for some subsequence, it follows by Fatou’s Lemma

\[

\int_0^1 |g|^2\, dx \le \liminf_k \int_0^1 |g_{k_\ell}|^2 \, dx \le n^2

\]

so $g \in B_n$.

As $L^2[0,1] = \bigcup_n B_n$ is a countable union of nowhere dense sets, it is of first category.

- Quotient topologies and equivalence classes
- Is the sum of factorials of first $n$ natural numbers ever a perfect cube?
- Integer solutions of $x^3+y^3=z^2$
- Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
- Rotating x,y points 45 degrees
- Proof for sequent (Logic)
- Free Group Generated By Image
- Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions
- $G/Z$ cannot be isomorphic to quaternion group
- Find $\lim_{x \to – \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$
- $PA^2\sin A+PB^2\sin B+PC^2\sin C$ is minimum if P is the incenter.
- Sum of two periodic functions
- Proving the so-called “Well Ordering Principle”
- Solve for $\theta$: $a = b\tan\theta – \frac{c}{\cos\theta}$
- How to prove this inequality(7)?