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 $0<A\leq B<\infty$ where for each $v\in V$ we have  

$$A\lVert v\rVert^2\leq \sum_{k\in\mathbb{N}}|\langle v,e_k\rangle|^2\leq B\lVert v\rVert^2\ \forall \ \ v\in V$$

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 $0<A\leq B<\infty$ such that for every $x\in\mathcal{H}$ we have

$$A\lVert x\rVert^2\leq \sum_i |\langle x,x_i\rangle|^2\leq B\lVert x\rVert^2$$,

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

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