Some thoughts:
Let \(\displaystyle{\left(X,||\cdot||_{X}\right)=\left(Y,||\cdot||_{Y}\right)=\left(\mathbb{R},\left|\cdot\right|\right)}\).
Consider \(\displaystyle{W\subseteq \mathbb{R}}\) a subspace of \(\displaystyle{\left(\mathbb{R},+,\cdot\right)}\) over \(\displaystyle{\mathbb{R}}\).
\(\displaystyle{\dim_{\mathbb{R}}\,W\leq 1\implies \dim_{\mathbb{R}}\,W=0\,\lor \dim_{\mathbb{R}}\,W=1\implies W=\left\{0\right\}\,\lor W=\mathbb{R}}\)
and thus, \(\displaystyle{\mathbb{R}}\) is the only dense subspace of \(\displaystyle{\mathbb{R}}\).
Let \(\displaystyle{f\in\,\mathbb{B}\,\left(\mathbb{R},\mathbb{R}\right)}\). Then, the real function \(\displaystyle{f}\) is continuous at \(\displaystyle{\mathbb{R}}\)
and satisfies the relations :
\(\displaystyle{f(x+y)=f(x)+f(y)\,,x\,,y\in\mathbb{R}\,\,,f(a\,x)=a\,f(x)\,,a\in\mathbb{R}\,,x\in\mathbb{R}}\). So,
\(\displaystyle{a\in\mathbb{R}\implies f(a)=f\left(a\cdot 1\right)=a\,f(1)}\).
Now, the functions \(\displaystyle{f_{c}:\mathbb{R}\longrightarrow \mathbb{R}\,,f_{c}(x)=c\,x\,,c\in\mathbb{R}}\) are continuous and linear and then :
\(\displaystyle{f_{c}\in \mathbb{B}\,\left(\mathbb{R},\mathbb{R}\right)\,,\forall\,c\in\mathbb{R}}\).
So, \(\displaystyle{\mathbb{B}\,\left(\mathbb{R},\mathbb{R}\right)=\left\{f_{c}\,,c\in\mathbb{R}\right\}}\)
If \(\displaystyle{f_{c}\in\,\mathbb{B}\,\left(\mathbb{R},\mathbb{R}\right)}\), then :
\(\displaystyle{||f||=\sup\,\left\{\left|f_{c}(x)\right|\,,x\in\mathbb{R}\,\left|x\right|\leq 1\right\}=\sup\,\left\{\left|c\right|\,\left|x\right|\,x\in\mathbb{R}\,\left|x\right|\leq 1\right\}=\left|c\right|}\) .
Let \(\displaystyle{\left(f_{n}\right)_{n\in\mathbb{N}}}\) be a sequence of \(\displaystyle{\left(\mathbb{B}\,\left(\mathbb{R},\mathbb{R}\right),||\cdot||\right)}\) .
Then, there exists real sequence \(\displaystyle{\left(c_{n}\right)_{n\in\mathbb{N}}}\) such that
\(\displaystyle{f_{n}(x)=c_{n}\,x\,,\forall\,n\in\mathbb{N}\,,\forall\,x\in\mathbb{R}}\).
Consider \(\displaystyle{\epsilon>0}\) . There is \(\displaystyle{n_{0}\in\mathbb{N}}\) such that \(\displaystyle{||f_{n}-f_{m}||<\epsilon\,,\forall\,n\,m\geq n_{0}}\).
Let \(\displaystyle{n\,,m\geq n_{0}}\). The function \(\displaystyle{f_{n}-f_{m}}\) is given by
\(\displaystyle{(f_{n}-f_{m})(x)=f_{n}(x)-f_{m}(x)=\left(c_{n}-c_{m}\right)\,x\,,x\in\mathbb{R}}\), so :
\(\displaystyle{||f_{n}-f_{m}||=\left|c_{n}-c_{m}\right|<\epsilon}\).
In conclusion, the real \(\displaystyle{\left(c_{n}\right)_{n\in\mathbb{N}}}\) is a Cauchy-sequence and since \(\displaystyle{\left(\mathbb{R},\left|\cdot\right|\right)}\)
we get : \(\displaystyle{c_{n}\to c\,,n\to \infty}\) for some \(\displaystyle{c\in\mathbb{R}}\). Let \(\displaystyle{\epsilon>0}\). There is \(\displaystyle{k\in\mathbb{N}}\)
such that \(\displaystyle{\left|c_{n}-c\right|<\epsilon\,,\forall\,n\geq n_{0}}\). We define \(\displaystyle{f:\mathbb{R}\longrightarrow \mathbb{R}\,,f(x)=c\,x}\)
and then \(\displaystyle{f\in \mathbb{B}\,(\mathbb{R},\mathbb{R})}\). Also, for every \(\displaystyle{n\geq n_{0}}\) :
\(\displaystyle{||f_{n}-f||=\left|c_{n}-c\right|<\epsilon}\), which means that \(\displaystyle{f_{n}\to f\,,n\to \infty}\) .
This result was expected because, if \(\displaystyle{\left(Y,||\cdot||\right)}\), is a \(\displaystyle{\rm{Banach}}\) space, then so is \(\displaystyle{\left(\mathbb{B}\,(X,Y),||\cdot||\right)}\).
Additional question :
If \(\displaystyle{f_{n}:\mathbb{R}\longrightarrow \mathbb{R}\,,f_{n}(x)=\dfrac{x}{2^{n}}\,,n\in\mathbb{N}}\), then :
calculate the series \(\displaystyle{\sum_{n=1}^{\infty}f_{n}}\).
Also, by defining \(\displaystyle{g:\mathbb{B}\,(\mathbb{R},\mathbb{R})\longrightarrow \mathbb{R}\equiv \mathbb{M}_{1}\,(\mathbb{R})\,,f_{c}\mapsto c}\)
we have that :
\(\displaystyle{g\,(f_{c_1}+f_{c_{2}})=g\,(f_{c_1+c_2})=c_1+c_2=g(f_{c_1})+g(f_{c_2})\,,\forall\,,c_1\,,c_2\in\mathbb{R}}\)
\(\displaystyle{g\,(a\,f_{c})=g\,(f_{a\,c})=a\,c=a\,g(f_{c})\,,\forall\,a\in\mathbb{R}\,,\forall\,c\in\mathbb{R}}\)
and obviously \(\displaystyle{g}\) is one-one, and onto \(\mathbb{R}\).
Futhermore, \(\displaystyle{||g\,(f_{c})||=\left|c\right|=||f_{c}||\,,\forall\,c\in\mathbb{R}}\) .
Therefore, the \(\displaystyle{\mathbb{R}}\) - spaces \(\displaystyle{\left(\mathbb{B}\,(\mathbb{R},\mathbb{R}),+,\cdot\right)\,,\left(\mathbb{R},+,\cdot\right)}\)
are isometrically isomorphic and then : \(\displaystyle{\mathbb{B}\,(\mathbb{R},\mathbb{R})=<g^{-1}(1)>=<Id_{\mathbb{R}}>}\). Indeed,
\(\displaystyle{Id_{\mathbb{R}}\neq \mathbb{O}}\) and if \(\displaystyle{f\in \mathbb{B}\,(\mathbb{R},\mathbb{R})}\), then \(\displaystyle{f=f_{c}}\)
for some \(\displaystyle{c\in\mathbb{R}}\) and
\(\displaystyle{f(x)=f_{c}(x)=c\,x=c\,Id_{\mathbb{R}}\,(x)\,\forall\,x\in\mathbb{R}\implies f=c\,Id_{\mathbb{R}}}\) .
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Functional analysis and Algebra