= Functional analysis
{wiki}
Functional analysis brings <linear algebra> and <topology> together to study <vector spaces> of <functions> and the <linear operators> acting on them. A <function space> such as $C([0,1])$ has infinitely many independent directions: the <monomials> $1,x,x^2,\ldots$ are <linearly independent>. Finite <linear combinations> alone cannot describe all its <continuous functions>. A useful theory must also say when approximations converge, whether their limits belong to the chosen space, and which operations preserve that <norm convergence>. <Normed vector spaces>, <Banach spaces>, and <bounded linear operators> provide these three controls.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/4ae9a5af8622dba3aa37461b71d08407_MIT18_102s21_lec1.pdf[MIT 18.102, lecture 1: the infinite-dimensional viewpoint]{ref}
A <bounded linear operator> $T:X\to Y$ between <normed vector spaces> satisfies $\|Tf\|_Y\leq M\|f\|_X$ for every $f$, with one finite constant $M$. The least possible constant is its <operator norm>. By <linearity>, the same estimate controls approximation errors:
$$
\|Tf-Tg\|_Y\leq\|T\|\,\|f-g\|_X.
$$
This is exactly <continuity> for a <linear map> between <normed vector spaces>; neither space needs to be complete. It turns an input accuracy guarantee into an output accuracy guarantee, even when the input has infinitely many degrees of freedom.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/22bf2f774fb161776032491568148707_MIT18_102s21_lec2.pdf[MIT 18.102, lecture 2: boundedness, continuity, and the operator norm]{ref}
For a concrete <linear functional> on $C([0,1])$, take the <integral> $F(f)=\int_0^1 f(x)\,dx$. In the <supremum norm>,
$$
|F(f)|\leq\int_0^1|f(x)|\,dx\leq\|f\|_\infty.
$$
The constant function $f=1$ attains equality, so $\|F\|=1$. For $f(x)=x^2$, the output is $F(f)=1/3$; any approximation $g$ with $\|g-x^2\|_\infty\leq0.01$ therefore has $|F(g)-1/3|\leq0.01$. The <Cauchy-Schwarz inequality> also gives $|F(f)|\leq\|f\|_2\|1\|_2=\|f\|_2$ on this interval, again with equality at $1$. Thus integration remains a <bounded linear functional> under the <L2 norm>, although <point evaluation> behaves very differently.
The beauty of this viewpoint is that one estimate can justify an entire approximation procedure. Once a <sequence> converges in the chosen <norm>, every <bounded linear operator> carries its limit along. <Completeness> ensures that Cauchy approximations actually describe an element of the space. The choice of <norm> matters: it determines which errors are small and which measurements are stable. The examples below distinguish an average, which ignores sufficiently narrow peaks, from <point evaluation>, which detects their full height. For applications, this decides whether reducing an overall fitting error guarantees an accurate reported measurement. A small <L2 norm> of the error controls its <integral> on this interval, while it gives no such guarantee for a single sampled value.
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