Is the closure of any linear subspace of a normed space $X$ again a linear subspace of $X$? Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$. I'm not able to visualize the additive and multiplicative closure of the new two points added to the subspace $\bar{A}$.
 A: It is sufficient to prove that $\alpha x + \beta y \in \overline{X}$ where $\alpha, \beta$ are in the underlying field $\mathbb{F}$ and $x, y \in \overline{X}$. We know that $0 \in \overline{X}$ since $X \subset \overline{X}$. Since $x, y \in \overline{X}$ there exist $x_{j}, y_{j} \in X$ such that $x_j \to x$ and $y_j \to y$. Since multiplication and addition are continuous $\alpha x_j + \beta y_j \to \alpha x + \beta y$. Therefore, $\alpha x + \beta y \in \overline{X}$
A: Show that the closure $\overline Y$ of a subspace $Y$ of a normed space $X$ is again a vector subspace.
Proof:
Suppose $x, y \in \overline Y$. There are three cases we must consider. 


*

*If $x, y \in Y$ then it is immediate that $x + y \in Y$ since $Y$ is a subspace of $X$ and thus $x + y \in \overline Y$.

*If $x \in \overline Y \setminus Y$ and $y \in Y$ then by definition, given $\varepsilon > 0$ there exists $x_0 \in Y$ with $x_0 \neq x$ such that $d(x_0, x) = \|x_0 - x\| < \varepsilon$. Now observe that since $x_0 \neq x$ then $x_0 + y \neq x + y$ and $x_0 + y \in Y$. Now we have $d(x_0 + y, x + y) = \|x_0 + y - x - y\| = \|x_0 - x\| < \varepsilon$. Hence we can conclude that $x_0 + y$ is an accumulation point of $Y$. Hence $x + y \in \overline Y$.

*If $x, y \in \overline Y \setminus Y$ then by definition given $\varepsilon > 0$ there exists $x_0 \in Y$ with $x_0 \neq x$ such that $d(x_0, x) = \|x_0 - x\| < \varepsilon/2$. Similarly, there exists $y_0 \in Y$ with $y_0 \neq y$ such that $d(y_0, y) = \|y_0 - y\| < \varepsilon/2$. If $x_0 + y_0 = x + y$ choose some $x_0 \neq x_0' \in Y$ such that $d(x_0', x) = \|x_0' - x\| < \varepsilon/2$ which is true since there are infinitely many elements in $Y$ with that property. So suppose without loss of generality that $x_0 + y_0 \neq x + y$. It follows that $$d(x_0 + y_0, x + y) = \|x_0 + y_0 - x - y\| \leq \|x_0 - x\| + \|y_0 - y\| < \frac\varepsilon2 + \frac\varepsilon2 = \varepsilon.$$ Hence we have shown that $x + y$ is an accumulation point of $Y$ and hence $x + y \in \overline Y$.


Now suppose $x \in \overline Y$. Suppose $\alpha$ is a scalar. We must consider two cases.


*

*If $x \in Y$ then since $Y$ is a subspace of $X$ it follows immediately that $\alpha x \in Y$ and thus $\alpha x \in \overline Y$.

*If $x \in \overline Y \setminus Y$ then $x$ is an accumulation point of $Y$. By definition given $\varepsilon > 0$ there exists $x_0 \in Y$ with $x_0 \neq x$ such that $d(x_0, x) = \|x_0 - x\| < \varepsilon/(\vert \alpha \vert + 1)$. Now since $x_0 \neq x$ then $\alpha x_0 \neq \alpha x$ and observe that $$d(\alpha x_0, \alpha x) = \|\alpha x_0 - \alpha x \| = \vert \alpha \vert \|x_0 - x\| < \vert \alpha \vert \frac{\varepsilon}{\vert \alpha \vert +1} < \varepsilon.$$ Thus we have shown that $\alpha x$ is an accumulation point and we can conclude that $\alpha x \in \overline Y$.


Thus we can conclude that $\overline Y$ is a subspace of $X$.
Note: It is my understanding that the closure of a set is the set along with all its limit points or accumulation points. It is my understanding that it NOT implied that all the elements of a set are limit points of the set. Please correct me if I'm wrong here!
A: Suppose $f,g\in \overline{A}$, $c\in F$, the underlying field, and let $\epsilon>0$. Then $f$ and $g$ must either be in $A$ or a limit point of $A$, so we can find $\hat f$ and $\hat g$ in $A$ so that $$\|f-\hat f\| < \frac{\epsilon}{2|c|},\qquad\|g-\hat g\| < \frac{\epsilon}{2}.$$ Then by the triangle inequality $$\|(cf+g)-(c\hat f + \hat g)\| \leq |c|\,\|f-\hat f\| + \|g-\hat g\| < \epsilon,$$ so $cf+g\in\overline{A}$. It follows that $\overline{A}$ is a subspace.
