A proper vector subspace of a topological vector space has no interior

Statement

Let $(X,\mathcal{J},$$\mathbb{K}$$)$ be a topological vector space and let $(Y,\mathbb{K})$ be a vector subspace of $(X,\mathbb{K})$ (so $Y\subseteq X$), then:

• If $Y$ is a proper vector subspace of $X$ then $\text{Int}$$(Y)\eq\emptyset$ - i.e. it has no interior, or empty interior.
  • Symbolically: $(Y\subset X)\implies \text{Int}(Y;X)\eq \emptyset$

Proof

Let $(X,\mathcal{J},\mathbb{K})$ and let $(Y,\mathbb{K})$ be a vector subspace of $(X,\mathbb{K})$, so $Y\subseteq X$. We have 2 cases:

1. Suppose $Y\eq X$ - by the nature of logical implication we do not care about the truth or falsity of the RHS and we're done
2. Suppose $Y\subset X$ - by the nature of logical implication we must show the RHS holds.
   • Recall: "For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal"
     • Symbolically: $(\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y$
   • Recall also that the contrapositive of an implication is logically equivalent to the original statement, that is:
     • $(A\implies B)\iff(\neg B\implies\neg A)$
   • So we use the earlier statement to see:
     • $\big[(\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y\big]$

       $\iff$

     $\big[X\neq Y\implies(\forall U\in(\mathcal{J}-\{\emptyset\})[U\nsubseteq Y])\big]$

     • See subset and negation of subset for information on what these mean.
   • We have $X\neq Y$ (as $Y\subset X$ is a proper subset there must be something in $X$ that is not in $Y$
     • So we see $X\neq Y$
       • $\implies \forall U\in(\mathcal{J}-\{\emptyset\})[U\nsubseteq Y]$ - in words, for all non-empty open sets of $X$, that open set is not contained in $Y$.
       • Recall the definition of interior:
         • $$\text{Int}(Y):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq Y\} } U$$
           • blah blah blah
             • We see $\text{Int}(Y)\eq\emptyset$

Grade: D

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