Definition :- Quotient Space

Hello friends, In this video we see the definition of Quotient Space.
Let N be a any vector space and M is subspace then Quotient Space N/M is defined as collection of all left cosets
N/M = { x+M/ x is in N}.
and this N/M (Quotient Space ) is Vector space. with binary operations
for x+M and y+M are in N/M and a in F
I) (x+M) + (y+M)= (x+y)+M
II) a(x+M) = ax+M
with zero element as M

and one thing is to be noted that
if x is in N and also this x is in M. then the left coset formed by this x is same as M
i.e. x is M then x+M = M ( proof is in video)

Quotient Space is normed linear space
Quotient Space is Banach space
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Quotient Space is Vector space

[Sakshi_Barman_]

Thank u so much sir, very good explanation