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Let $\mathbb M$ be linearly closed subset ( $\forall\ x, y \in M: \alpha x + \beta y \in M$) of vector topological space $(\mathbb L, \tau)$. I am wondering under which conditions $\mathbb M$ would be topological vector space itself in topology induced by $\tau$. My idea was:

For a linearly closed subset $\mathbb M$ of vector topological space $(\mathbb L, \tau)$ to have induced topology consistent with the linear structure it is necessary and sufficient for $\mathbb M$ to be closed in initial topology $\tau$ of $\mathbb L$.

While trying to prove necessity I came up with a statement that if $\mathbb M$ is linearly closed it implies induced topology being consistent with the linear structure on $\mathbb M$. At the same time, I suppose that consistency alone isn't sufficient for $\mathbb M$ to be linearly closed. But so far I failed with proving implication ( $\mathbb M$ is linearly closed $\Rightarrow$ $\mathbb M$ is closed in initial topology $\tau$ of $\mathbb L$ ). So is my initial statement correct?

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You are right that closedness often occurs in this type of statement, but in this case the answer is: $\mathbb{M}$ is ALWAYS a topological vector space with the induced topology. In my opinion, this accounts for some of the beauty of functional analysis.

Proof: we have to show that vector addition and scalar multiplication are continuous and that $\mathbb{M}$ is Hausdorff. Addition is a function $$+_\mathbb{M}:\mathbb{M}\times\mathbb{M}\to\mathbb{M}$$ which is (general property of initial topology) continuous iff $$\iota\circ+_\mathbb{M}:\mathbb{M}\times\mathbb{M}\to\mathbb{L}$$ is continuous, where $\iota$ is the embedding $\mathbb{M}\to\mathbb{L}$. But this function is equal to $$+_\mathbb{L}\circ(\iota\times\iota):\mathbb{M}\times\mathbb{M}\to\mathbb{L}.$$ Now, $$(\iota\times\iota):\mathbb{M}\times\mathbb{M}\to\mathbb{L}\times\mathbb{L}$$ is continuous, since (again general property, this time in $\mathbb{L}\times\mathbb{L}$, whose topology is induced by the projections $\pi_{1,2}: \mathbb{L}\times\mathbb{L}\to\mathbb{L}$) $$\pi_{1,2}\circ(\iota\times\iota):\mathbb{M}\times\mathbb{M}\to\mathbb{L}$$ is equal to $$\iota\circ\pi_{1,2}: \mathbb{M}\times\mathbb{M}\to\mathbb{L}$$ which is continuous as a composition of continuous functions (they are the ones w.r.t. which the topologies are formed). Hence, $$\iota\circ+_\mathbb{M}:\mathbb{M}\times\mathbb{M}\to\mathbb{L}$$ from above is continuous, again as a composition. Continuity of scalar multiplication is proved similarly.

The fact that $\mathbb{M}$ is Hausdorff is generally true for subspaces of Hausdorff spaces.

I hope you can understand my proof, don't hesitate to ask! It's really not that difficult, the notational hassle is just terrible...

Btw: a much more general statement is also true, namely that vector spaces together with an initial topology w.r.t. a family of separating, linear (!) transformations are TVS; the proof is quite similar.

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