Intereting Posts

invariance of integrals for homotopy equivalent spaces
The “Empty Tuple” or “0-Tuple”: Its Definition and Properties
Is there a nice way to classify the ideals of the ring of lower triangular matrices?
Checking flat- and smoothness: enough to check on closed points?
Agreement of $q$-expansion of modular forms
Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.
Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$?
If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
Does $\pi_1(X, x_0)$ act on $\tilde{X}$?
Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
Characterisation of one-dimensional Sobolev space
Is $|\mathbb{R}$| = |$\mathbb{R^2}$| = … = |$\mathbb{R^\infty}$|?
How many solutions are there to $x_1 + x_2 + … + x_5 = 21$?
Foundations of logic
Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries.

For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by:

$$

\displaystyle\|A\|_1=\max_{1\leq j\leq n}\sum_{i=1}^{n}|a_{ij}|;

$$

$$

\displaystyle\|A\|_\infty=\max_{1\leq i\leq n}\sum_{j=1}^{n}|a_{ij}|;

$$

$$

\displaystyle\|A\|_\text{max}=\max\{|a_{ij}|\}.

$$

Matrix $A\in \mathbb{R}^{n\times n}$ is said to be positive definite iff

$$

\langle Ax, x\rangle> 0 \quad \forall x\in\mathbb{R}^n\setminus\{0\}.

$$

Let $S$ be the set of all positive definite matrices on $\mathbb{R}^{n\times n}$. Prove that $S$ is an open set in $(X,\|.\|_1)$,

$(X,\|.\|_\infty$), $(X,\|.\|_\text{max})$.

**I would like to thank all for their help and comments.**

- Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?
- Is duality an exact functor on Banach spaces or Hilbert spaces?
- Metrizability of weak convergence by the bounded Lipschitz metric
- The weak$^*$ topology on $X^*$ is not first countable if $X$ has uncountable dimension.
- Convergence on Norm vector space.
- Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

- To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$?
- Matrix proof using norms
- Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?
- What is the difference between a Hamel basis and a Schauder basis?
- $\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system
- Bounded linear operator maps norm-bounded, closed sets to closed sets. Implies closed range?
- Is Banach-Alaoglu equivalent to AC?
- Norm of the integral operator in $L^2(\mathbb{R})$.
- When does pointwise convergence imply uniform convergence?
- Non brute force proof of multiplicative property of determinants

Restricting to the unit ball is always illustrating. Let $A$ be a given positive definite matrix, then there is $\delta>0$ such that \begin{equation}

<Ax,x>\ge\delta

\end{equation} for all $\|x\|=1$.

We use the 2-norm, defined by \begin{equation}

\|A\|=\operatorname{sup}_{\|x\|=1}\|Ax\|,

\end{equation} which is equivalent to any other norms.

If $B$ is very close to $A$, say, $\|B-A\|<\epsilon$, then

\begin{equation}

|<Bx,x>-<Ax,x>|=|<(B-A)x,x>|<\epsilon\|x\|^2,

\end{equation} so if you restrict to the unit ball again then you can bound $<Bx,x>$ from below using positive definiteness of $A$ and controlling $\epsilon$, and this will lead to the positive definiteness of $B$.

- Present a function with specific feature
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- Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n – 1} \le n$