The definition of a finite frame and putting equations in blogger

Okay here is an attempt to writing an equation... I used an exchange post to modify the html

Example 1: $x^2$

Example 2: $$x^2$$ 

This was the link I used: https://support.google.com/blogger/thread/234344761/inserting-latex-equations?hl=en

DEFINITION (FRAME) Let $V$ be an inner product space $\{e_k{\}}_{k\in \mathbb{N}}$ be a set of vectors in $V$. The set $\{e_k{\}}_{k\in \mathbb{N}}$ is a frame of $V$ if it satisfies the frame condition. This is, if there exists constants $A,B$ such that $0<A\le B<\mathrm{\infty }$ where for each $v\in V$ we have  

$$A\Vert v{\Vert }^2\le \sum _{k\in \mathbb{N}}|⟨v,e_k⟩{|}^2\le B\Vert v{\Vert }^2\mathrm{\forall }v\in V$$

where $A,B$ are the frame bounds.

This is the other definition I used from the book "Frames for undergraduates"...

DEFINITION (FRAME) A frame for a Hilbert space $\mathcal{H}$ is a sequence of vectors $\{x_i\}\subset \mathcal{H}$ for which there exist constant $0<A\le B<\mathrm{\infty }$ such that for every $x\in \mathcal{H}$ we have

$$A\Vert x{\Vert }^2\le \sum _i|⟨x,x_i⟩{|}^2\le B\Vert x{\Vert }^2$$,

where again $A,B$ are the frame bounds. I will be following notation used in this definition for most of my posts.