Hello, I encountered a closed set for a certain operation and I didnt completely understand the concept or I think I understand but I need someone to confirm. So in example I will highlight, I'm proving something in LA and using a property of a vector subspace being "closed for addition and multiplication". Similar thing is also mentioned in the core definition of a vector subspace. What I think this means: A set closed for some operation means that using that operation within a set will still give us an element that is within that set. For example, lets say vectors u and v are from U. Then u+v would also be from U because U is closed for addition. Another example for multiplication: If we have vector u from U and a scalar alpha, then alpha*u will also be from U because U is closed for multiplication. Just want to have this sorted out. (I didn't want to use latex because it's a bigger text and there's not much to write when it comes to actual mathematics notation)
#Need clarification on what exactly ''closed set'' is
mint temple
jaunty jacinthBOT
Hello, I encountered a closed set for a certain operation and I didnt completely...
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gloomy sun
Hello, I encountered a closed set for a certain operation and I didnt completely...
that's completely right!
it's also important to look at the domain, like scalar multiplication in a vectorspace
$\mathbb{K} \times V \to V$
bright gardenBOT
Djake3tooth
gloomy sun
if we say V is closed under scalar multiplication
we mean that for any $k \in \mathbb{K}, v \in V$ we have $k \cdot v \in V$
bright gardenBOT
Djake3tooth
mint temple
we mean that for any $k \in \mathbb{K}, v \in V$ we have $k \cdot v \in V$
yeah, thanks!
muted falconBOT
@mint temple has given 1 rep to @gloomy sun
pure ocean
With the title I literally thought it was the definition of a topologically closed set
gloomy sun
With the title I literally thought it was the definition of a topologically clos...
well closed set in topology is a set closed under limit-of-a-sequence operation so
it fits this definition
pure ocean
Yes i see
mint temple
Okay I got another proof thats rather confusing, I dont want to shit around with making another post so I will post here:
Operations with vector subspaces is theme
Let U and W be vector subspaces of V. If UuW (union) is also a subspace of V, then U is a subset of W or W is a subset of U.
I dont last part at all
why cant they be finitely small subspaces
why does one need to be a subset of the other is my question to be precise
pure ocean
if V U W is a subspace then for v in V and u in W u is in V U W and v also but that means that u+v is in V U W so either in V or W if it’s in V then u+v -v is in V thus u is in V so that means W C V and if u+v is in W then likewise v is in W so V C W