= Coercive bilinear form
{title2=$B(v,v)\geq\alpha\|v\|^2,\quad\alpha>0$}
= Coercive
{synonym}
A real <bilinear form> $B$ on a <normed vector space> $X$ is coercive when there is one constant $\alpha>0$ such that
$$
B(v,v)\geq\alpha\|v\|^2\qquad(v\in X).
$$
The lower bound uniformly controls every direction in the chosen <norm>. Unlike the upper bound defining a <bounded bilinear form>, it forces positive diagonal values at every nonzero <vector>. No symmetry is assumed. For complex <Hilbert spaces>, the usual analogue uses a <sesquilinear form> and $\operatorname{Re}B(v,v)\geq\alpha\|v\|^2$; it is not the same assertion for a complex <bilinear form>.
A concrete bounded and coercive example on $\mathbb R^2$, with the <Euclidean norm>, is
$$
A=\begin{pmatrix}2&1\\-1&3\end{pmatrix},\qquad B(u,v)=u^TAv.
$$
Its <quadratic form> is
$$
q(x,y)=B((x,y),(x,y))=2x^2+3y^2\geq2(x^2+y^2).
$$
The best lower constant is $\alpha=2$, attained at $(1,0)$. Nevertheless, $B(e_1,e_2)=1$ and $B(e_2,e_1)=-1$, so coercivity does not imply a <symmetric bilinear form>. The <symmetric part of a matrix> $S=(A+A^T)/2=\operatorname{diag}(2,3)$ determines $q$, while the <skew-symmetric matrix> part changes mixed pairings without changing $q$. This is the <symmetric and alternating parts of a bilinear form> decomposition.<Bilinear Forms by Keith Conrad>[Theorem 1.7 and equation (1.3): symmetric and alternating parts]{ref}
For the upper constant, direct multiplication gives $A^TA=\begin{pmatrix}5&-1\\-1&10\end{pmatrix}$. Its <eigenvalues> are $(15\pm\sqrt{29})/2$, so the optimal upper constant, the <spectral norm>, is $C=\sqrt{(15+\sqrt{29})/2}$. The two constants answer different questions: how large a mixed pairing can be, and how small a self-pairing can be relative to squared <norm>.
\Image[/bilinear-form-coercivity-energy-landscapes.png]
{title=Bounded saddle and flat-direction quadratic forms contrasted with a uniformly coercive bowl}
The plotted <quadratic forms> show the mechanism: $x^2-y^2$ descends in one direction, $x^2$ leaves one direction uncontrolled, and $2x^2+3y^2$ controls both. These are self-pairings; their <level sets> cannot reveal the <skew-symmetric matrix> part. Calling $q$ a quadratic energy here refers to this diagnostic quantity, not to a claim that a nonsymmetric equation comes from minimizing an <energy functional>.
On a real <finite-dimensional vector space> with any <norm>, strict positivity $B(v,v)>0$ for every nonzero $v$ does imply coercivity. Indeed the <quadratic form> is <continuous>, and its positive minimum on the <compact> <unit sphere> supplies $\alpha$. If $B$ is also a <symmetric bilinear form>, this is <positive definiteness>. For general $B$, it is the <symmetric and alternating parts of a bilinear form> decomposition that isolates the positive part. Switching between <equivalent norms> preserves coercivity, but usually changes its constant.
The <infinite-dimensional vector space> distinction is explicit on the real <l2 sequence space>:
$$
B(u,v)=\sum_{n=1}^{\infty}\frac{u_nv_n}{n}.
$$
The <Cauchy-Schwarz inequality> gives boundedness with $C=1$, and $B(v,v)>0$ for every nonzero <vector>. But $B(e_n,e_n)=1/n$ with $\|e_n\|_2=1$, so no positive $\alpha$ works. On the <span> of $e_1,\ldots,e_N$, the best lower constant is $1/N$: each finite restriction is coercive, but the control disappears as $N$ grows. Strict positivity is therefore weaker than a uniform lower bound.
The practical payoff is a stability estimate for a <weak formulation>. Assume $H$ is a real <Hilbert space>, $B:H\times H\to\mathbb R$ is both a <bounded bilinear form> and a <coercive bilinear form>, and the load $\ell\in H'$ is a <bounded linear functional>. The <Lax-Milgram theorem> gives a unique $u\in H$ satisfying $B(u,v)=\ell(v)$ for all $v\in H$, including nonsymmetric $B$.<Introduction to Finite Element Methods by Christian Clason>[§2.2, Theorem 2.7, pp. 20–21, equation (2.9): hypotheses, unique solvability and load bound]{ref} The <dual norm> $\|\ell\|_{H'}=\sup_{v\ne0}|\ell(v)|/\|v\|$ measures load size. Testing with $v=u$ gives the short <energy estimate>
$$
\alpha\|u\|^2\leq B(u,u)=\ell(u)\leq\|\ell\|_{H'}\|u\|,\qquad
\|u\|\leq\frac{\|\ell\|_{H'}}{\alpha}.
$$
For $u=0$ the bound holds directly; otherwise divide by $\|u\|$. Subtracting the equations for two loads and using the same argument gives $\|u_1-u_2\|\leq\|\ell_1-\ell_2\|_{H'}/\alpha$. Thus a positive lower constant limits the response to load errors. The upper bound and <completeness> of the <Hilbert space> enter the existence theorem; the estimate itself uses the lower bound and the load's <dual norm>.
For the displayed <matrix> $A$, take $\ell(v)=f^Tv$. Then $B(u,v)=\ell(v)$ for every $v$ means $A^Tu=f$, not $Au=f$, with this ordering of the <bilinear form>. Since $\det A=7$,
$$
u=\frac17\begin{pmatrix}3&1\\-1&2\end{pmatrix}f,\qquad
\|u\|_2\leq\frac12\|f\|_2.
$$
For $f=t(1,0)^T$, the exact response is $u=t(3,-1)^T/7$ and $\|u\|_2=\sqrt{10}|t|/7<|t|/2$. For $f=t(0,1)^T$, it is $u=t(1,2)^T/7$, with <norm> $\sqrt5|t|/7$. The actual maximal amplification is the <operator norm> $\|(A^T)^{-1}\|_2=\sqrt{2/(15-\sqrt{29})}<1/2$: the coercivity estimate is valid but need not be sharp for a nonsymmetric <bilinear form>.
\Image[/coercive-bilinear-form-load-response-bound.png]
{title=Solutions for unit loads lie inside the coercivity bound, and response grows at most linearly with load norm}
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