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I used $GL(V)=: \{V\in \mathbb{C}^n : \exists W\in \mathbb{C}^n, VW = WV = 1\}$, which is to express

enter image description here

as highlighted.

But it sounds having errors, although it can compile. But I want to know is there anything that I can make it better.

Thank you!

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    For one thing, the spacing won’t be correct unless you put the entire expression in a single $...$ or \(...\) block. – Davislor Sep 6 '20 at 17:25
  • Hello, @Davislor, sounds better, thanks:-) – Frances H. Sep 6 '20 at 17:30
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    Welcome to the site! – Davislor Sep 6 '20 at 17:32
  • It’s ambiguous in this sample whether GL(V) is the product of a variable G and a function L, or a function named GL. In the latter case, you might want to format it differently, e.g. \DeclareMathOperator{\GL}{GL} or \newcommand{\GL}{\mathop{\textnormal{\slshape GL}}}. – Davislor Sep 6 '20 at 17:36
  • I'm not understanding what you mean by "sounds having errors" and "sounds better". – Teepeemm Sep 6 '20 at 19:14
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I add my proposal using simply \mathsf{GL} (sans-serif font) for GL.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools,amssymb}
\begin{document}
$\mathsf{GL}(\mathrm{V})\eqqcolon\{\mathrm{V}\in \mathbb{C}^n \colon \exists \mathrm{W}\in \mathbb{C}^n, \mathrm{VW} = \mathrm{WV} = 1\}$
\end{document}

enter image description here

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    If you choose to render V in upright lettering toward the end of the formula, you should probably give it the same typographic treatment in GL(V). – Mico Sep 16 '20 at 19:30
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    @Mico It is absolutely true :-) I provide to fix it. – Sebastiano Sep 16 '20 at 19:33
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You wrote,

I want to know is there anything that I can [do to] make it better

If the GL term is an operator, it's best to typeset it using upright roman letters. Separately, I think the overall look would improve if slightly greater visual prominence were given to the curly braces, e.g., by making them \big.

enter image description here

\documentclass{article}
\usepackage{amssymb} % for '\mathbb' macro
\usepackage{amsmath} % for '\DeclareMathOperator' macro
\DeclareMathOperator{\GL}{GL}
\usepackage{mathtools} % for '\eqqcolon' macro
\begin{document}
\[
\GL(V) \eqqcolon \bigl\{ V\in \mathbb{C}^n : \exists W\in \mathbb{C}^n, VW = WV = 1 \bigr\}
\]
\end{document}

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